<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">R</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>set </b><font color="Maroon">IR</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">CL</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x</font>;<br/>
<a NAME="E1:50"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font> implies for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:50_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 ;<br/>

<a NAME="E3:50_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/>
<a NAME="E4:50_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<b>hence </b><a NAME="E5:50_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="neckla_3.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:50"/>
( ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> ) ) implies <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> )
 ;<br/>

<a NAME="E2:50_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:50_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:50_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="neckla_3.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E3:50_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/>
<a NAME="E4:50_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<b>hence </b><a NAME="E5:50_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:50_2"/>
 <a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:50_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:50_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x</font>
 
;<br/>
<a NAME="E3:50_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E4:50_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/>
<b>hence </b><a NAME="E5:50_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="neckla_3.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:50_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:50"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:50"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font> iff for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> ) )
 <b>by </b><i>, </i>;<br/>


</div>
