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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( (  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">PA</font> &amp; <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">PA</font> or  not <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) ) implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) );<br/>
<a NAME="E1:53_1_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  ex <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_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="E1:53_1_1_1_1"/>
(  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">e</font>,<font color="Maroon">PA</font> &amp; <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) or for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">e</font>,<font color="Maroon">PA</font> or  not <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_1"/>
 ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">e</font>,<font color="Maroon">PA</font> &amp; <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1_1_1"/>
 ex <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Olive">u</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_2"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">e</font>,<font color="Maroon">PA</font> or  not <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1_1_2"/>
 ex <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Olive">u</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Y</font>, <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E3:53_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <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>  <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <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>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E6:53_1_1"/>
for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds <br/> ( (  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Olive">y</font>,<font color="Maroon">PA</font> &amp; <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Olive">y</font>,<font color="Maroon">PA</font> or  not <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) ) implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) )
 <b>by </b><i/>;<br/>


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