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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of  <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">F</font> holds  <font color="Olive" title="b2">f</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Olive" title="b3">g</font></span><br/><b>let </b><font color="Maroon" title="c2">F</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of  <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">F</font> holds  <font color="Olive" title="b1">f</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Olive" title="b2">g</font></span><br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c4">g</font></span><br/>





<a NAME="E1:136"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">F</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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>)
 
<b>by </b><i><a class="ref" href="monoid_0.html#T25" target="_self" title="MONOID_0:th.25">Th25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">f'</font> = <font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c6">g'</font> = <font color="Maroon" title="c4">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> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:136"/>
<font color="Maroon" title="c3">f</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f'</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">g'</font>
 
<b>by </b><i><a class="ref" href="monoid_0.html#T27" target="_self" title="MONOID_0:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E4:136"/>
<font color="Maroon" title="c3">f</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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