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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> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="filter_0.html#K9" target="_self">&lt;=&gt;</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E2:83_1_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) )
 <b>from </b><i><a class="ref" href="relat_1.html#S1">RELAT_1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E3:83_1_1"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:83_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="wellset1.html#T1">WELLSET1:1</a></i>;<br/>
<b>hence </b><a NAME="E3:83_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="filter_0.html#E2:83_1_1"><i><font color="Green">E57</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E4:83_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="filter_0.html#K9" target="_self">&lt;=&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:83_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:83_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="filter_0.html#K9" target="_self">&lt;=&gt;</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="filter_0.html#E2:83_1_1"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:83_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="filter_0.html#K9" target="_self">&lt;=&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E5:83_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="filter_0.html#K9" target="_self">&lt;=&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> implies <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="filter_0.html#E2:83_1_1"><i><font color="Green">E57</font></i></a></i>;<br/>


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