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<b>thus </b><a NAME="E1:1_1_1"/>
( <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> implies for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>  st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<b>thus </b><a NAME="E2:1_1_1"/>
( ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>  st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) implies <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>  st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a><br/>

<b>assume </b><a NAME="E2:1_1_1_1"/><i><font color="Green">E42</font></i>: 
<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> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E3:1_1_1_1"/><b>then </b>
 ex <font color="Olive">x1</font>, <font color="Olive">y1</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">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">y1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/>
<b>hence </b><a NAME="E4:1_1_1_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> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/>


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


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