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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">g</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">op</font> = <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font>);<br/>
<a NAME="E2:144"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c2">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font></span>)</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>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> &amp; <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">j</font> = <font color="Maroon" title="c5">h</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:144"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="partfun1.html#T120" title="PARTFUN1:th.120">PARTFUN1:120</a></span>;<br/><span class="kw">thus </span><font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">h</font> = 
<font color="Maroon" title="c4">f</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">h</font>

<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c6">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></span>
<br/>.= 
<font color="Maroon" title="c5">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T37" title="PARTFUN1:th.37">PARTFUN1:37</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>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font></span><br/><span class="kw">thus </span><font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c4">f</font> = 
<font color="Maroon" title="c5">h</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">f</font>

<br/>.= 
<font color="Maroon" title="c6">j</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c2">g</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></span>
<br/>.= 
<font color="Maroon" title="c5">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T36" title="PARTFUN1:th.36">PARTFUN1:36</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="E5:144"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</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>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</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/>
<span class="kw">hence </span><a NAME="E6:144"/>
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</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>; <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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