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<html><head><meta charset="utf-8"><title>Algebra.Properties.Group</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Comment">------------------------------------------------------------------------</a>
<a id="74" class="Comment">-- The Agda standard library</a>
<a id="103" class="Comment">--</a>
<a id="106" class="Comment">-- Some derivable properties</a>
<a id="135" class="Comment">------------------------------------------------------------------------</a>
<a id="209" class="Symbol">{-#</a> <a id="213" class="Keyword">OPTIONS</a> <a id="221" class="Pragma">--cubical-compatible</a> <a id="242" class="Pragma">--safe</a> <a id="249" class="Symbol">#-}</a>
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<a id="1486" href="Algebra.Properties.Group.html#1458" class="Function">∙-cancel</a> <a id="1495" class="Symbol">=</a> <a id="1497" href="Algebra.Properties.Group.html#1077" class="Function">∙-cancelˡ</a> <a id="1507" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="1509" href="Algebra.Properties.Group.html#1273" class="Function">∙-cancelʳ</a>
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<a id="1722" href="Algebra.Properties.Group.html#1568" class="Bound">x</a> <a id="1743" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
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<a id="1791" href="Algebra.Properties.Group.html#1746" class="Function">⁻¹-injective</a> <a id="1804" class="Symbol">{</a><a id="1805" href="Algebra.Properties.Group.html#1805" class="Bound">x</a><a id="1806" class="Symbol">}</a> <a id="1808" class="Symbol">{</a><a id="1809" href="Algebra.Properties.Group.html#1809" class="Bound">y</a><a id="1810" class="Symbol">}</a> <a id="1812" href="Algebra.Properties.Group.html#1812" class="Bound">eq</a> <a id="1815" class="Symbol">=</a> <a id="1817" href="Algebra.Properties.Group.html#1273" class="Function">∙-cancelʳ</a> <a id="1827" href="Algebra.Properties.Group.html#1805" class="Bound">x</a> <a id="1829" href="Algebra.Properties.Group.html#1809" class="Bound">y</a> <a id="1831" class="Symbol">(</a> <a id="1833" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="1841" href="Algebra.Properties.Group.html#1805" class="Bound">x</a> <a id="1843" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="1845" href="Algebra.Properties.Group.html#1805" class="Bound">x</a> <a id="1847" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="1850" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1853" href="Algebra.Structures.html#4748" class="Function">inverseʳ</a> <a id="1862" href="Algebra.Properties.Group.html#1805" class="Bound">x</a> <a id="1864" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1868" href="Algebra.Bundles.html#7485" class="Field">ε</a> <a id="1877" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="1881" href="Algebra.Structures.html#4748" class="Function">inverseʳ</a> <a id="1890" href="Algebra.Properties.Group.html#1809" class="Bound">y</a> <a id="1892" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="1896" href="Algebra.Properties.Group.html#1809" class="Bound">y</a> <a id="1898" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="1900" href="Algebra.Properties.Group.html#1809" class="Bound">y</a> <a id="1902" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="1905" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="1909" href="Algebra.Structures.html#1369" class="Function">∙-congˡ</a> <a id="1917" href="Algebra.Properties.Group.html#1812" class="Bound">eq</a> <a id="1920" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="1924" href="Algebra.Properties.Group.html#1809" class="Bound">y</a> <a id="1926" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="1928" href="Algebra.Properties.Group.html#1805" class="Bound">x</a> <a id="1930" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="1933" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a> <a id="1935" class="Symbol">)</a>
<a id="⁻¹-anti-homo-∙"></a><a id="1938" href="Algebra.Properties.Group.html#1938" class="Function">⁻¹-anti-homo-∙</a> <a id="1953" class="Symbol">:</a> <a id="1955" class="Symbol">∀</a> <a id="1957" href="Algebra.Properties.Group.html#1957" class="Bound">x</a> <a id="1959" href="Algebra.Properties.Group.html#1959" class="Bound">y</a> <a id="1961" class="Symbol">→</a> <a id="1963" class="Symbol">(</a><a id="1964" href="Algebra.Properties.Group.html#1957" class="Bound">x</a> <a id="1966" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="1968" href="Algebra.Properties.Group.html#1959" class="Bound">y</a><a id="1969" class="Symbol">)</a> <a id="1971" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="1974" href="Algebra.Bundles.html#7431" class="Field Operator">≈</a> <a id="1976" href="Algebra.Properties.Group.html#1959" class="Bound">y</a> <a id="1978" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="1981" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="1983" href="Algebra.Properties.Group.html#1957" class="Bound">x</a> <a id="1985" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a>
<a id="1988" href="Algebra.Properties.Group.html#1938" class="Function">⁻¹-anti-homo-∙</a> <a id="2003" href="Algebra.Properties.Group.html#2003" class="Bound">x</a> <a id="2005" href="Algebra.Properties.Group.html#2005" class="Bound">y</a> <a id="2007" class="Symbol">=</a> <a id="2009" href="Algebra.Properties.Group.html#1077" class="Function">∙-cancelˡ</a> <a id="2019" class="Symbol">_</a> <a id="2021" class="Symbol">(</a> <a id="2023" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="2031" href="Algebra.Properties.Group.html#2003" class="Bound">x</a> <a id="2033" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2035" href="Algebra.Properties.Group.html#2005" class="Bound">y</a> <a id="2037" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2039" class="Symbol">(</a><a id="2040" href="Algebra.Properties.Group.html#2003" class="Bound">x</a> <a id="2042" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2044" href="Algebra.Properties.Group.html#2005" class="Bound">y</a><a id="2045" class="Symbol">)</a> <a id="2047" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="2053" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2056" href="Algebra.Structures.html#4748" class="Function">inverseʳ</a> <a id="2065" class="Symbol">_</a> <a id="2067" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2071" href="Algebra.Bundles.html#7485" class="Field">ε</a> <a id="2093" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="2097" href="Algebra.Structures.html#4748" class="Function">inverseʳ</a> <a id="2106" class="Symbol">_</a> <a id="2108" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="2112" href="Algebra.Properties.Group.html#2003" class="Bound">x</a> <a id="2114" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2116" href="Algebra.Properties.Group.html#2003" class="Bound">x</a> <a id="2118" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="2134" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2137" href="Algebra.Structures.html#1430" class="Function">∙-congʳ</a> <a id="2145" class="Symbol">(</a><a id="2146" href="Algebra.Properties.Group.html#621" class="Function">left-helper</a> <a id="2158" href="Algebra.Properties.Group.html#2003" class="Bound">x</a> <a id="2160" href="Algebra.Properties.Group.html#2005" class="Bound">y</a><a id="2161" class="Symbol">)</a> <a id="2163" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2167" class="Symbol">(</a><a id="2168" href="Algebra.Properties.Group.html#2003" class="Bound">x</a> <a id="2170" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2172" href="Algebra.Properties.Group.html#2005" class="Bound">y</a><a id="2173" class="Symbol">)</a> <a id="2175" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2177" href="Algebra.Properties.Group.html#2005" class="Bound">y</a> <a id="2179" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="2182" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2184" href="Algebra.Properties.Group.html#2003" class="Bound">x</a> <a id="2186" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="2189" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2192" href="Algebra.Structures.html#1876" class="Function">assoc</a> <a id="2198" class="Symbol">(</a><a id="2199" href="Algebra.Properties.Group.html#2003" class="Bound">x</a> <a id="2201" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2203" href="Algebra.Properties.Group.html#2005" class="Bound">y</a><a id="2204" class="Symbol">)</a> <a id="2206" class="Symbol">(</a><a id="2207" href="Algebra.Properties.Group.html#2005" class="Bound">y</a> <a id="2209" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a><a id="2211" class="Symbol">)</a> <a id="2213" class="Symbol">(</a><a id="2214" href="Algebra.Properties.Group.html#2003" class="Bound">x</a> <a id="2216" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a><a id="2218" class="Symbol">)</a> <a id="2220" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="2330" href="Algebra.Properties.Group.html#2313" class="Bound">x</a> <a id="2345" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2348" href="Algebra.Properties.Group.html#621" class="Function">left-helper</a> <a id="2360" href="Algebra.Properties.Group.html#2313" class="Bound">x</a> <a id="2362" href="Algebra.Properties.Group.html#2315" class="Bound">y</a> <a id="2364" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="2406" href="Algebra.Properties.Group.html#2315" class="Bound">y</a> <a id="2409" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2411" href="Algebra.Properties.Group.html#2315" class="Bound">y</a> <a id="2413" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="2416" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2419" href="Algebra.Structures.html#4748" class="Function">inverseʳ</a> <a id="2428" href="Algebra.Properties.Group.html#2315" class="Bound">y</a> <a id="2430" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2434" href="Algebra.Bundles.html#7485" class="Field">ε</a> <a id="2449" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="identityʳ-unique"></a><a id="2452" href="Algebra.Properties.Group.html#2452" class="Function">identityʳ-unique</a> <a id="2469" class="Symbol">:</a> <a id="2471" class="Symbol">∀</a> <a id="2473" href="Algebra.Properties.Group.html#2473" class="Bound">x</a> <a id="2475" href="Algebra.Properties.Group.html#2475" class="Bound">y</a> <a id="2477" class="Symbol">→</a> <a id="2479" href="Algebra.Properties.Group.html#2473" class="Bound">x</a> <a id="2481" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2483" href="Algebra.Properties.Group.html#2475" class="Bound">y</a> <a id="2485" href="Algebra.Bundles.html#7431" class="Field Operator">≈</a> <a id="2487" href="Algebra.Properties.Group.html#2473" class="Bound">x</a> <a id="2489" class="Symbol">→</a> <a id="2491" href="Algebra.Properties.Group.html#2475" class="Bound">y</a> <a id="2493" href="Algebra.Bundles.html#7431" class="Field Operator">≈</a> <a id="2495" href="Algebra.Bundles.html#7485" class="Field">ε</a>
<a id="2497" href="Algebra.Properties.Group.html#2452" class="Function">identityʳ-unique</a> <a id="2514" href="Algebra.Properties.Group.html#2514" class="Bound">x</a> <a id="2516" href="Algebra.Properties.Group.html#2516" class="Bound">y</a> <a id="2518" href="Algebra.Properties.Group.html#2518" class="Bound">eq</a> <a id="2521" class="Symbol">=</a> <a id="2523" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="2531" href="Algebra.Properties.Group.html#2516" class="Bound">y</a> <a id="2546" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2549" href="Algebra.Properties.Group.html#852" class="Function">right-helper</a> <a id="2562" href="Algebra.Properties.Group.html#2514" class="Bound">x</a> <a id="2564" href="Algebra.Properties.Group.html#2516" class="Bound">y</a> <a id="2566" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2570" href="Algebra.Properties.Group.html#2514" class="Bound">x</a> <a id="2572" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="2575" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2577" class="Symbol">(</a><a id="2578" href="Algebra.Properties.Group.html#2514" class="Bound">x</a> <a id="2580" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2582" href="Algebra.Properties.Group.html#2516" class="Bound">y</a><a id="2583" class="Symbol">)</a> <a id="2585" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2588" href="Relation.Binary.Structures.html#1577" class="Function">refl</a> <a id="2593" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="2595" href="Algebra.Structures.html#1219" class="Function">∙-cong</a> <a id="2602" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="2604" href="Algebra.Properties.Group.html#2518" class="Bound">eq</a> <a id="2607" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2611" href="Algebra.Properties.Group.html#2514" class="Bound">x</a> <a id="2613" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="2616" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2619" href="Algebra.Properties.Group.html#2514" class="Bound">x</a> <a id="2626" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2629" href="Algebra.Structures.html#4685" class="Function">inverseˡ</a> <a id="2638" href="Algebra.Properties.Group.html#2514" class="Bound">x</a> <a id="2640" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2644" href="Algebra.Bundles.html#7485" class="Field">ε</a> <a id="2659" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="identity-unique"></a><a id="2662" href="Algebra.Properties.Group.html#2662" class="Function">identity-unique</a> <a id="2678" class="Symbol">:</a> <a id="2680" class="Symbol">∀</a> <a id="2682" class="Symbol">{</a><a id="2683" href="Algebra.Properties.Group.html#2683" class="Bound">x</a><a id="2684" class="Symbol">}</a> <a id="2686" class="Symbol">→</a> <a id="2688" href="Algebra.Definitions.html#1425" class="Function">Identity</a> <a id="2697" href="Algebra.Properties.Group.html#2683" class="Bound">x</a> <a id="2699" href="Algebra.Bundles.html#7459" class="Field Operator">_∙_</a> <a id="2703" class="Symbol">→</a> <a id="2705" href="Algebra.Properties.Group.html#2683" class="Bound">x</a> <a id="2707" href="Algebra.Bundles.html#7431" class="Field Operator">≈</a> <a id="2709" href="Algebra.Bundles.html#7485" class="Field">ε</a>
<a id="2711" href="Algebra.Properties.Group.html#2662" class="Function">identity-unique</a> <a id="2727" class="Symbol">{</a><a id="2728" href="Algebra.Properties.Group.html#2728" class="Bound">x</a><a id="2729" class="Symbol">}</a> <a id="2731" href="Algebra.Properties.Group.html#2731" class="Bound">id</a> <a id="2734" class="Symbol">=</a> <a id="2736" href="Algebra.Properties.Group.html#2251" class="Function">identityˡ-unique</a> <a id="2753" href="Algebra.Properties.Group.html#2728" class="Bound">x</a> <a id="2755" href="Algebra.Properties.Group.html#2728" class="Bound">x</a> <a id="2757" class="Symbol">(</a><a id="2758" href="Agda.Builtin.Sigma.html#246" class="Field">proj₂</a> <a id="2764" href="Algebra.Properties.Group.html#2731" class="Bound">id</a> <a id="2767" href="Algebra.Properties.Group.html#2728" class="Bound">x</a><a id="2768" class="Symbol">)</a>
<a id="inverseˡ-unique"></a><a id="2771" href="Algebra.Properties.Group.html#2771" class="Function">inverseˡ-unique</a> <a id="2787" class="Symbol">:</a> <a id="2789" class="Symbol">∀</a> <a id="2791" href="Algebra.Properties.Group.html#2791" class="Bound">x</a> <a id="2793" href="Algebra.Properties.Group.html#2793" class="Bound">y</a> <a id="2795" class="Symbol">→</a> <a id="2797" href="Algebra.Properties.Group.html#2791" class="Bound">x</a> <a id="2799" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2801" href="Algebra.Properties.Group.html#2793" class="Bound">y</a> <a id="2803" href="Algebra.Bundles.html#7431" class="Field Operator">≈</a> <a id="2805" href="Algebra.Bundles.html#7485" class="Field">ε</a> <a id="2807" class="Symbol">→</a> <a id="2809" href="Algebra.Properties.Group.html#2791" class="Bound">x</a> <a id="2811" href="Algebra.Bundles.html#7431" class="Field Operator">≈</a> <a id="2813" href="Algebra.Properties.Group.html#2793" class="Bound">y</a> <a id="2815" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a>
<a id="2818" href="Algebra.Properties.Group.html#2771" class="Function">inverseˡ-unique</a> <a id="2834" href="Algebra.Properties.Group.html#2834" class="Bound">x</a> <a id="2836" href="Algebra.Properties.Group.html#2836" class="Bound">y</a> <a id="2838" href="Algebra.Properties.Group.html#2838" class="Bound">eq</a> <a id="2841" class="Symbol">=</a> <a id="2843" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="2851" href="Algebra.Properties.Group.html#2834" class="Bound">x</a> <a id="2866" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2869" href="Algebra.Properties.Group.html#621" class="Function">left-helper</a> <a id="2881" href="Algebra.Properties.Group.html#2834" class="Bound">x</a> <a id="2883" href="Algebra.Properties.Group.html#2836" class="Bound">y</a> <a id="2885" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2889" class="Symbol">(</a><a id="2890" href="Algebra.Properties.Group.html#2834" class="Bound">x</a> <a id="2892" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2894" href="Algebra.Properties.Group.html#2836" class="Bound">y</a><a id="2895" class="Symbol">)</a> <a id="2897" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2899" href="Algebra.Properties.Group.html#2836" class="Bound">y</a> <a id="2901" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="2904" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2907" href="Algebra.Structures.html#1430" class="Function">∙-congʳ</a> <a id="2915" href="Algebra.Properties.Group.html#2838" class="Bound">eq</a> <a id="2918" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2927" href="Algebra.Bundles.html#7485" class="Field">ε</a> <a id="2930" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="2932" href="Algebra.Properties.Group.html#2836" class="Bound">y</a> <a id="2934" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="2937" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2940" href="Algebra.Structures.html#2972" class="Function">identityˡ</a> <a id="2950" class="Symbol">(</a><a id="2951" href="Algebra.Properties.Group.html#2836" class="Bound">y</a> <a id="2953" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a><a id="2955" class="Symbol">)</a> <a id="2957" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2971" href="Algebra.Properties.Group.html#2836" class="Bound">y</a> <a id="2973" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="2976" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="inverseʳ-unique"></a><a id="2979" href="Algebra.Properties.Group.html#2979" class="Function">inverseʳ-unique</a> <a id="2995" class="Symbol">:</a> <a id="2997" class="Symbol">∀</a> <a id="2999" href="Algebra.Properties.Group.html#2999" class="Bound">x</a> <a id="3001" href="Algebra.Properties.Group.html#3001" class="Bound">y</a> <a id="3003" class="Symbol">→</a> <a id="3005" href="Algebra.Properties.Group.html#2999" class="Bound">x</a> <a id="3007" href="Algebra.Bundles.html#7459" class="Field Operator">∙</a> <a id="3009" href="Algebra.Properties.Group.html#3001" class="Bound">y</a> <a id="3011" href="Algebra.Bundles.html#7431" class="Field Operator">≈</a> <a id="3013" href="Algebra.Bundles.html#7485" class="Field">ε</a> <a id="3015" class="Symbol">→</a> <a id="3017" href="Algebra.Properties.Group.html#3001" class="Bound">y</a> <a id="3019" href="Algebra.Bundles.html#7431" class="Field Operator">≈</a> <a id="3021" href="Algebra.Properties.Group.html#2999" class="Bound">x</a> <a id="3023" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a>
<a id="3026" href="Algebra.Properties.Group.html#2979" class="Function">inverseʳ-unique</a> <a id="3042" href="Algebra.Properties.Group.html#3042" class="Bound">x</a> <a id="3044" href="Algebra.Properties.Group.html#3044" class="Bound">y</a> <a id="3046" href="Algebra.Properties.Group.html#3046" class="Bound">eq</a> <a id="3049" class="Symbol">=</a> <a id="3051" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="3059" href="Algebra.Properties.Group.html#3044" class="Bound">y</a> <a id="3067" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="3070" href="Relation.Binary.Structures.html#1603" class="Function">sym</a> <a id="3074" class="Symbol">(</a><a id="3075" href="Algebra.Properties.Group.html#1520" class="Function">⁻¹-involutive</a> <a id="3089" href="Algebra.Properties.Group.html#3044" class="Bound">y</a><a id="3090" class="Symbol">)</a> <a id="3092" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="3096" href="Algebra.Properties.Group.html#3044" class="Bound">y</a> <a id="3098" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="3101" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="3104" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="3107" href="Algebra.Structures.html#4571" class="Function">⁻¹-cong</a> <a id="3115" class="Symbol">(</a><a id="3116" href="Relation.Binary.Structures.html#1603" class="Function">sym</a> <a id="3120" class="Symbol">(</a><a id="3121" href="Algebra.Properties.Group.html#2771" class="Function">inverseˡ-unique</a> <a id="3137" href="Algebra.Properties.Group.html#3042" class="Bound">x</a> <a id="3139" href="Algebra.Properties.Group.html#3044" class="Bound">y</a> <a id="3141" href="Algebra.Properties.Group.html#3046" class="Bound">eq</a><a id="3143" class="Symbol">))</a> <a id="3146" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="3150" href="Algebra.Properties.Group.html#3042" class="Bound">x</a> <a id="3152" href="Algebra.Bundles.html#7507" class="Field Operator">⁻¹</a> <a id="3158" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="3162" class="Comment">------------------------------------------------------------------------</a>
<a id="3235" class="Comment">-- DEPRECATED NAMES</a>
<a id="3255" class="Comment">------------------------------------------------------------------------</a>
<a id="3328" class="Comment">-- Please use the new names as continuing support for the old names is</a>
<a id="3399" class="Comment">-- not guaranteed.</a>
<a id="3419" class="Comment">-- Version 1.1</a>
<a id="left-identity-unique"></a><a id="3435" href="Algebra.Properties.Group.html#3435" class="Function">left-identity-unique</a> <a id="3456" class="Symbol">=</a> <a id="3458" href="Algebra.Properties.Group.html#2251" class="Function">identityˡ-unique</a>
<a id="3475" class="Symbol">{-#</a> <a id="3479" class="Keyword">WARNING_ON_USAGE</a> <a id="3496" class="Pragma">left-identity-unique</a>
<a id="3517" class="String">&quot;Warning: left-identity-unique was deprecated in v1.1.
Please use identityˡ-unique instead.&quot;</a>
<a id="3610" class="Symbol">#-}</a>
<a id="right-identity-unique"></a><a id="3614" href="Algebra.Properties.Group.html#3614" class="Function">right-identity-unique</a> <a id="3636" class="Symbol">=</a> <a id="3638" href="Algebra.Properties.Group.html#2452" class="Function">identityʳ-unique</a>
<a id="3655" class="Symbol">{-#</a> <a id="3659" class="Keyword">WARNING_ON_USAGE</a> <a id="3676" class="Pragma">right-identity-unique</a>
<a id="3698" class="String">&quot;Warning: right-identity-unique was deprecated in v1.1.
Please use identityʳ-unique instead.&quot;</a>
<a id="3792" class="Symbol">#-}</a>
<a id="left-inverse-unique"></a><a id="3796" href="Algebra.Properties.Group.html#3796" class="Function">left-inverse-unique</a> <a id="3816" class="Symbol">=</a> <a id="3818" href="Algebra.Properties.Group.html#2771" class="Function">inverseˡ-unique</a>
<a id="3834" class="Symbol">{-#</a> <a id="3838" class="Keyword">WARNING_ON_USAGE</a> <a id="3855" class="Pragma">left-inverse-unique</a>
<a id="3875" class="String">&quot;Warning: left-inverse-unique was deprecated in v1.1.
Please use inverseˡ-unique instead.&quot;</a>
<a id="3966" class="Symbol">#-}</a>
<a id="right-inverse-unique"></a><a id="3970" href="Algebra.Properties.Group.html#3970" class="Function">right-inverse-unique</a> <a id="3991" class="Symbol">=</a> <a id="3993" href="Algebra.Properties.Group.html#2979" class="Function">inverseʳ-unique</a>
<a id="4009" class="Symbol">{-#</a> <a id="4013" class="Keyword">WARNING_ON_USAGE</a> <a id="4030" class="Pragma">right-inverse-unique</a>
<a id="4051" class="String">&quot;Warning: right-inverse-unique was deprecated in v1.1.
Please use inverseʳ-unique instead.&quot;</a>
<a id="4143" class="Symbol">#-}</a>
</pre></body></html>
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