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<span class="kw">set </span><font color="Maroon" title="c2">G</font> =  <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">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="c4">D</font> =  <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>);<br/>
<a NAME="E2:133_1_1"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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#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/>
<div><span class="lab"><font color="Green" title="E62">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">f9</font>, <font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c7">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="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f9</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f9</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c7">h</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">fg</font> = <font color="Maroon" title="c5">f9</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c9">gh</font> = <font color="Maroon" title="c6">g</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c7">h</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> ;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">f9</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c7">h</font> = 
<font color="Maroon" title="c7">h</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c8">fg</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="c7">h</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c5">f9</font></span>)</span>
<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="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c5">f9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>
<br/>.= 
<font color="Maroon" title="c9">gh</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c5">f9</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">f9</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c7">h</font></span>)</span>
<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>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c5">f1</font> = <font color="Maroon" title="c3">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="txt" href="#E2:133_1_1"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="partfun1.html#T45" title="PARTFUN1:th.45">PARTFUN1:45</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f1</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font> &amp; <font color="Maroon" title="c6">h</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">j</font> = <font color="Maroon" title="c6">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="#E2:133_1_1"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="partfun1.html#T46" title="PARTFUN1:th.46">PARTFUN1:46</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c5">f1</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">h</font> = 
<font color="Maroon" title="c7">j</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c3">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="c6">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T6" title="PARTFUN1:th.6">PARTFUN1:6</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">h</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c6">h</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">f1</font> = 
<font color="Maroon" title="c3">g</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c7">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="c6">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T7" title="PARTFUN1:th.7">PARTFUN1:7</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:133_1_1"/>
(  <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  &amp;  <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp;  not  <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133_1_1"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E3:133_1_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="monoid_0.html#E28"><span class="lab"><font color="Green" title="E15">Lm2</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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