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<div class="add">

<b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">Y</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> ;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> ;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Olive">x'</font> where <font color="Olive">x'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x'</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E3:66"/>
for <font color="Olive">F</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">F</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K23" target="_self">GPart</a> <font color="Maroon">a</font> holds <br/>for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> holds <br/><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K23" target="_self">GPart</a> <font color="Maroon">a</font>
 ;<br/>

<b>thus </b><a NAME="E2:66_1"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:66_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1_1_1"/>
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> or <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1_1"><b>case </b></a><a NAME="E1:66_1_1_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:66_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">x5</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:66_1_1_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x5</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E6:66_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:66_1_1_1_1_1"><i><font color="Green">E29</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1_2"><b>case </b></a><a NAME="E1:66_1_1_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:66_1_1_1_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">x5</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:66_1_1_1_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x5</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E6:66_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:66_1_1"/>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:66"/>
<font color="Maroon">a</font> <a href="bvfunc_1.html#R2" target="_self">is_dependent_of</a>  <a href="bvfunc_1.html#K23" target="_self">GPart</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div>
