<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  implies ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  iff ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  &amp; the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</a>  ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:32"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  iff ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  &amp; the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:32"/>
( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  implies ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</a>  ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  &amp; the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c2">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>
<span class="kw">assume </span><a NAME="E2:32_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</a>  )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E4:32_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c2">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T12" target="_self" title="MONOID_0:th.12">Th12</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E25">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> )</span><br/><a NAME="E1:32_1_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
 ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c2">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Olive" title="b2">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="monoid_0.html#T12" target="_self" title="MONOID_0:th.12">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E2:32_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E5:32_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="monoid_0.html#E28"><span class="lab"><font color="Green" title="E15">Lm2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font></span><br/><span class="kw">thus </span><a NAME="E3:32_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E4:32_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E1:32_1_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="monoid_0.html#E28"><span class="lab"><font color="Green" title="E15">Lm2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A7</font></span>: 
( <font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E8:32_1"/>
<font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="monoid_0.html#T6" target="_self" title="MONOID_0:th.6">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</a> </span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>] <span class="kw">means </span>( $1 <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; $2 <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> );<br/>
<a NAME="E9:32_1"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">b</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E2:32_1_2"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c7">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="monoid_0.html#T12" target="_self" title="MONOID_0:th.12">Th12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>]</span><br/>

<a NAME="E3:32_1_2"/>
( <font color="Maroon" title="c7">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E6:32_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E7:32_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="monoid_0.html#E28"><span class="lab"><font color="Green" title="E15">Lm2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:32_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E2:32_1_2"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:32_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <br/>for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E9:32_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">u</font> being   <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>)<span class="kw"> such that </span><br/><a NAME="E12:32_1"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>]
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )</span><br/><a NAME="E1:32_1_3"/>
( <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c3">x</font> )
 ;<br/><span class="kw">hence </span><a NAME="E2:32_1_3"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E7:32_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E14:32_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></span>;<br/>
<a NAME="E15:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">u</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="finseqop.html#D2" title="FINSEQOP:def.2">FINSEQOP:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">u</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="finseqop.html#D1" title="FINSEQOP:def.1">FINSEQOP:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) )</span><br/>

<a NAME="E16:32_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E17:32_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E15:32_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:32"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D10" target="_self" title="MONOID_0:def.10">MONOID_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</a>  or <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c2">u</font> being   <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>)<span class="kw"> such that </span><a NAME="E5:32"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
<font color="Maroon" title="c2">u</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>)
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="finseqop.html#D2" title="FINSEQOP:def.2">FINSEQOP:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D5" target="_self" title="MONOID_0:def.5">MONOID_0:def 5</a>,<a class="ref" href="monoid_0.html#D16" target="_self" title="MONOID_0:def.16">MONOID_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b2">r</font>, <font color="Olive" title="b3">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <br/>( the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b3">l</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <br/>( the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">l</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )</span><br/>



<span class="kw">take </span><font color="Maroon" title="c5">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <br/>( the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">l</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c6">r</font> = <font color="Maroon" title="c4">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )</span><br/>

<span class="kw">thus </span><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">l</font> = 
<font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">l</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="monoid_0.html#E28"><span class="lab"><font color="Green" title="E15">Lm2</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>)</span>)</span>,<font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="ref" href="finseqop.html#D1" title="FINSEQOP:def.1">FINSEQOP:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="setwiseo.html#T23" title="SETWISEO:th.23">SETWISEO:23</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font></span><br/>

<span class="kw">thus </span><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c3">a</font> = 
<font color="Maroon" title="c6">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a</font>

<br/>.= 
<font color="Maroon" title="c4">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="monoid_0.html#E28"><span class="lab"><font color="Green" title="E15">Lm2</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="ref" href="finseqop.html#D1" title="FINSEQOP:def.1">FINSEQOP:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="setwiseo.html#T23" title="SETWISEO:th.23">SETWISEO:23</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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