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<b>let </b><font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c2">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">R</font> be    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c1">E</font>;<br/><b>let </b><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<b>thus </b><a NAME="E1:46"/>
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">R</font> implies  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:46_1"/><i><font color="Green" title="E9">A1</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> <b>by </b><i><a class="txt" href="relset_1.html#E3:46_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">a</font>
;<br/>

<b>thus </b><a NAME="E5:46_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font>
 <b>by </b><i><a class="txt" href="relset_1.html#E3:46_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font><b> such that </b><a NAME="E2:46"/><i><font color="Green" title="E9">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font>
 ;<br/>

<b>thus </b><a NAME="E3:46"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">R</font>
 <b>by </b><i><a class="txt" href="relset_1.html#E2:46"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="relat_1.html#T20" title="RELAT_1:th.20">RELAT_1:20</a></i>;<br/>


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