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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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp;  ex <font color="Olive" title="b1">S</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] ) );<br/>
<a NAME="E1:3_1"/><i><font color="Green" title="E3">A1</font></i>: 
for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">t</font>, <font color="Olive" title="b3">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">t</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive" title="b3">v</font>] holds <br/><font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c1">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:3_1"/><i><font color="Green" title="E4">A2</font></i>: 
for <font color="Olive" title="b1">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> iff  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">t</font>] ) )
 <b>from </b><i><a class="ref" href="tarski.html#S1" title="TARSKI:sch.1">TARSKI:sch 1</a>(<a class="txt" href="wellset1.html#E1:3_1"><i><font color="Green" title="E3">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c1">B</font>
;<br/>

<b>let </b><font color="Maroon" title="c2">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/>

<a NAME="E4:3_1"/><i><font color="Green" title="E5">A3</font></i>: 
( <font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> iff  ex <font color="Olive" title="b1">T</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <font color="Olive" title="b1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">R</font>] ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:3_1_1"/>
( <font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> implies  ex <font color="Olive" title="b1">T</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <font color="Olive" title="b1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">R</font>] ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_1_1"/>
<font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:3_1_1_1"/><i><font color="Green" title="E6">A4</font></i>: 
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">R</font> &amp;  ex <font color="Olive" title="b1">S</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] ) )
 <b>by </b><i><a class="txt" href="wellset1.html#E3:3_1"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">y</font> as   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> <b>by </b><i><a class="txt" href="wellset1.html#E3:3_1_1_1"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">y</font>
;<br/>

<b>thus </b><a NAME="E5:3_1_1_1"/>
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">R</font>] )
 <b>by </b><i><a class="txt" href="wellset1.html#E3:3_1_1_1"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:3_1_1"/>
(  ex <font color="Olive" title="b1">T</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <font color="Olive" title="b1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">R</font>] ) implies <font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E3:3_1"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:3_1"/>
( <font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> implies ( <font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">R</font>] ) )
 ;<br/>

<b>thus </b><a NAME="E6:3_1"/>
( <font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">R</font>] implies <font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E4:3_1"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/>


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