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<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:142"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:142"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">Y</font>,<font color="Maroon">Y</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>
 
<b>by </b><i><a class="ref" href="partfun1.html#T128">PARTFUN1:128</a></i>;<br/>
<a NAME="E3:142"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">Y</font>,<font color="Maroon">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">Y</font>,<font color="Maroon">Y</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E4:142"/>
( <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K13" target="_self">GPFuncs</a> <font color="Maroon">Y</font>) <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">Y</font>,<font color="Maroon">Y</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></i>;<br/>
<a NAME="E5:142"/><b>then </b><i><font color="Green">E40</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K13" target="_self">GPFuncs</a> <font color="Maroon">Y</font>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">Y</font>,<font color="Maroon">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">Y</font>,<font color="Maroon">Y</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K13" target="_self">GPFuncs</a> <font color="Maroon">X</font>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="142_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:142_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">Y</font>,<font color="Maroon">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">Y</font>,<font color="Maroon">Y</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:142_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">Y</font>,<font color="Maroon">Y</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">Y</font>,<font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">x2</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K13" target="_self">GPFuncs</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i><a class="ref" href="monoid_0.html#T16" target="_self">Th16</a></i>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">y2</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="struct_0.html#NM1">Element</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> <b>by </b><i>, <a class="txt" href="monoid_0.html#E41"><i><font color="Green">Lemma70</font></i></a>, <a class="ref" href="monoid_0.html#T16" target="_self">Th16</a></i>;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K13" target="_self">GPFuncs</a> <font color="Maroon">Y</font>) <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">x1</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">x2</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E40"><i><font color="Green">Lemma69</font></i></a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>
<br/>.= 
<font color="Maroon">x1</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">x2</font>
<b>by </b><i><a class="ref" href="monoid_0.html#T16" target="_self">Th16</a></i>
<br/>.= 
<font color="Maroon">y1</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="monoid_0.html#T16" target="_self">Th16</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K13" target="_self">GPFuncs</a> <font color="Maroon">X</font>) <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E40"><i><font color="Green">Lemma69</font></i></a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:142"/>
<font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K13" target="_self">GPFuncs</a> <font color="Maroon">Y</font>) <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K13" target="_self">GPFuncs</a> <font color="Maroon">X</font>)
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D23" target="_self">MONOID_0:def 23</a><br/>


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