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<b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E2:144_1_1"/>
( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> &amp;  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K13" title="MONOID_0:func.13">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> )
 
<b>by </b><i><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a>, <a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a>, <a class="ref" href="funct_2.html#T141" title="FUNCT_2:th.141">FUNCT_2:141</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">F</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K13" title="MONOID_0:func.13">GPFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<a NAME="E4:144_1_1"/><i><font color="Green" title="E67">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">F</font> holds  <font color="Olive" title="b1">x</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">F</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c5">y</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="monoid_0.html#E129"><i><font color="Green" title="E66">Lm9</font></i></a></i>;<br/>
<a NAME="E2:144_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">g</font>
 
<b>by </b><i><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></i>;<br/>
<b>hence </b><a NAME="E3:144_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">G</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of  <a href="monoid_0.html#K13" title="MONOID_0:func.13">GPFuncs</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:144_1_1"/><i><font color="Green" title="E68">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font>
 <b>from </b><i><a class="ref" href="monoid_0.html#S1" target="_self" title="MONOID_0:sch.1">MONOID_0:sch 1</a>(<a class="txt" href="monoid_0.html#E4:144_1_1"><i><font color="Green" title="E67">A1</font></i></a>);<br/></i>
<a NAME="E7:144_1_1"/>
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">G</font> = <font color="Maroon" title="c4">G</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of  <a href="monoid_0.html#K13" title="MONOID_0:func.13">GPFuncs</a> <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="monoid_0.html#E6:144_1_1"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="monoid_0.html#T80" target="_self" title="MONOID_0:th.80">Th80</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">G</font>
;<br/>

<b>thus </b><a NAME="E9:144_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E6:144_1_1"><i><font color="Green" title="E68">A2</font></i></a></i>;<br/>


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