<?xml version="1.0"?>
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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:114_1"/>
not <!--HUMANRECFAILED-->( not <!--HUMANRECFAILED-->( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1"><b>suppose </b></a><a NAME="E1:114_1_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:114_1_1"/><b>then </b>
(  <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="partfun1.html#T172">PARTFUN1:172</a></i>;<br/><b>hence </b><a NAME="E3:114_1_1"/>
 <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2"><b>suppose </b></a><a NAME="E1:114_1_2"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:114_1_2"/>
for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3">:&gt;</a></span> iff <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1_2_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:114_1_2_1"/>
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3">:&gt;</a></span> implies <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:114_1_2_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3">:&gt;</a></span>
 ;<br/>

<a NAME="E2:114_1_2_1_1"/><b>then </b>
<font color="Maroon">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:114_1_2_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:114_1_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 ;<br/>

<a NAME="E3:114_1_2_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6" target="_self">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E5:114_1_2_1"/>
( <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3">:&gt;</a></span> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">g'</font> &amp; <font color="Maroon">g'</font> is <a href="partfun1.html#V1">total</a> )
 
<b>by </b><i>, , , <a class="ref" href="partfun1.html#T142">PARTFUN1:142</a></i>;<br/>
<b>hence </b><a NAME="E6:114_1_2_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3">:&gt;</a></span>
 <b>by </b><i><a class="ref" href="partfun1.html#D7">PARTFUN1:def 7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:114_1_2"/>
 <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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