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

<b>let </b><font color="Maroon">G</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/>

<b>assume </b><a NAME="E1:30"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V4">associative</a>
 ;<br/>

<b>thus </b><a NAME="E2:30"/>
( <font color="Maroon">G</font> is <a href="monoid_0.html#V13" target="_self">invertible</a> implies ( <font color="Maroon">G</font> is <a href="group_1.html#V2">unital</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="finseqop.html#NR2" target="_self">has_an_inverseOp</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">G</font> is <a href="monoid_0.html#V13" target="_self">invertible</a>
 ;<br/>

<b>consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:30_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:30_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E3:30_1_1"/>
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="monoid_0.html#D1" target="_self">Def1</a>, </i>;<br/><b>thus </b><a NAME="E4:30_1_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="monoid_0.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E6:30_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
;<br/>
<b>hence </b><a NAME="E7:30_1"/>
<font color="Maroon">G</font> is <a href="group_1.html#V2">unital</a>
 <b>by </b><i>, </i>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> );<br/>
<a NAME="E8:30_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>

<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:30_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<a NAME="E3:30_1_2"/>
( <font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, , <a class="ref" href="monoid_0.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E4:30_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>]
 <b>by </b><i><a class="ref" href="monoid_0.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:30_1"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> st <br/>for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>then consider </b><font color="Maroon">u</font> being   <a href="binop_1.html#NM1">UnOp</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>)<b> such that </b><br/><a NAME="E11:30_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>]
 ;<br/>
<b>take </b>
<font color="Maroon">u</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="finseqop.html#D2">FINSEQOP:def 2</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="finseqop.html#D1">FINSEQOP:def 1</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><a NAME="E1:30_1_3"/>
( <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">x</font> )
 ;<br/><b>hence </b><a NAME="E2:30_1_3"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="monoid_0.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:30_1"/><b>then </b>
<font color="Maroon">x</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>)
 
<b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>
<a NAME="E14:30_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E15:30_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) )
 <b>by </b><i><a class="ref" href="monoid_0.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:30"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D10" target="_self">MONOID_0:def 10</a><br/>

<b>given </b><font color="Maroon">u</font> being   <a href="binop_1.html#NM1">UnOp</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>)<b> such that </b><a NAME="E4:30"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">u</font> <a href="finseqop.html#R1">is_an_inverseOp_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>)
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="finseqop.html#D2">FINSEQOP:def 2</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D5" target="_self">MONOID_0:def 5</a>,<a class="ref" href="monoid_0.html#D16" target="_self">MONOID_0:def 16</a><br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>



<b>take </b><font color="Maroon">l</font> = <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font>;<br/>

<b>take </b><font color="Maroon">r</font> = <font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>;<br/>

<b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">l</font> = 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">l</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>)</span>)</span>,<font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="finseqop.html#D1">FINSEQOP:def 1</a></i>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="setwiseo.html#T23">SETWISEO:23</a></i>
;<br/>

<b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">r</font>,<font color="Maroon">a</font> = 
<font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>

<br/>.= 
<font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>)</span>)</span>
<b>by </b><i>, <a class="ref" href="finseqop.html#D1">FINSEQOP:def 1</a></i>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="setwiseo.html#T23">SETWISEO:23</a></i>
;<br/>


</div>
