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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">s</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">s</font> );<br/>
<a NAME="E1:28_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
;<br/>
<b>consider </b><font color="Maroon">CX</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font> iff  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="tarski.html#S1">TARSKI:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">CX</font>
;<br/>

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

<a NAME="E4:28_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font> iff  ex <font color="Olive">s</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">s</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:28_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font> implies  ex <font color="Olive">s</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">s</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_1_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  ex <font color="Olive">s</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">s</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E5:28_1_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="orders_3.html#E2"><i><font color="Green">Lemma17</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>thus </b><a NAME="E6:28_1_1_1_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="orders_3.html#E2"><i><font color="Green">Lemma17</font></i></a>, </i>;<br/>


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

<b>given </b><font color="Maroon">s</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><a NAME="E2:28_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s</font> )
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:28_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="orders_3.html#E2"><i><font color="Green">Lemma17</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:28_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:28_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font> iff  ex <font color="Olive">s</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">s</font> ) )
 ;<br/>


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