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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s1</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <font color="Maroon" title="c3">E</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font> );<br/>
<b>let </b><font color="Maroon" title="c5">X</font>, <font color="Maroon" title="c6">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <font color="Maroon" title="c3">E</font>,<span class="p3">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:33_1_2"/><i><font color="Green" title="E16">A8</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 ;<br/>

<b>assume </b><a NAME="E2:33_1_2"/><i><font color="Green" title="E17">A9</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 ;<br/>

<a NAME="E3:33_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_2_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 ;<br/><a NAME="E2:33_1_2_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>]
 <b>by </b><i><a class="txt" href="osalg_4.html#E1:33_1_2"><i><font color="Green" title="E16">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_2_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E2:33_1_2"><i><font color="Green" title="E17">A9</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:33_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 ;<br/>

<a NAME="E3:33_1_2_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>]
 
<b>by </b><i><a class="txt" href="osalg_4.html#E2:33_1_2"><i><font color="Green" title="E17">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:33_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E1:33_1_2"><i><font color="Green" title="E16">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:33_1_2"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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