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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">R</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Olive">R</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">R</font> );<br/>
<a NAME="E1:11_1_1"/><i><font color="Green">E30</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>proof </b></a><div class="add">

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

<b>given </b><font color="Maroon">R1</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><a NAME="E1:11_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R1</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:11_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R1</font>
 ;<br/>

<b>given </b><font color="Maroon">R2</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><a NAME="E3:11_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R2</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:11_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R2</font>
 ;<br/>

<a NAME="E5:11_1_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R2</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R2</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:11_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:11_1_1"/><i><font color="Green">E36</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">X</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>  <a href="classes2.html#K13">FinSETS</a>  &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">X</font>
;<br/>

<a NAME="E4:11_1_1"/>
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">X</font> iff  ex <font color="Olive">R</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">R</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:11_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies  ex <font color="Olive">R</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">R</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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="E3:11_1_1_2_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  &amp;  ex <font color="Olive">R</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Olive">R</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">R</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">R</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E5:11_1_1_2_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:11_1_1_2_1"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:11_1_1_2_1"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E8:11_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="classes2.html#K13">FinSETS</a> 
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E9:11_1_1_2_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="classes2.html#K13">FinSETS</a> 
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E10:11_1_1_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:11_1_1_2_1"/>
 ex <font color="Olive">R</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">R</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  )
 <b>by </b><i>, </i>;<br/>


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

<b>given </b><font color="Maroon">R</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><a NAME="E2:11_1_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">R</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E4:11_1_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E5:11_1_1_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:11_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 
<b>by </b><i>, <a class="ref" href="classes2.html#T64">CLASSES2:64</a></i>;<br/>
<b>hence </b><a NAME="E7:11_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:11_1_1"/>
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">X</font> iff  ex <font color="Olive">R</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Olive">R</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  ) )
 ;<br/>


</div>
