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<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<a NAME="E2:144_1_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> &amp;  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K13" target="_self">GPFuncs</a> <font color="Maroon">X</font>) <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="monoid_0.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_2.html#T12">FUNCT_2:12</a>, <a class="ref" href="funct_2.html#T141">FUNCT_2:141</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> =  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K13" target="_self">GPFuncs</a> <font color="Maroon">X</font></span>)</span> ;<br/>
<a NAME="E4:144_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font> holds  <font color="Olive">x</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font>;<br/>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font>, <font color="Maroon">g</font> = <font color="Maroon">y</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E2:144_1_1_1"/>
<font color="Maroon">x</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="monoid_0.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E3:144_1_1_1"/>
<font color="Maroon">x</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T12">FUNCT_2:12</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</font> being  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a>  <a href="monoid_0.html#M2" target="_self">SubStr</a> of  <a href="monoid_0.html#K13" target="_self">GPFuncs</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:144_1_1"/><i><font color="Green">E39</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>from </b><i><a class="ref" href="monoid_0.html#S1" target="_self">MONOID_0:sch 1</a>();<br/></i>
<a NAME="E7:144_1_1"/>
 <a href="partfun1.html#K6">id</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T12">FUNCT_2:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a> <a href="group_1.html#V2">unital</a>  <a href="monoid_0.html#M2" target="_self">SubStr</a> of  <a href="monoid_0.html#K13" target="_self">GPFuncs</a> <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E9:144_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


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