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<b>let </b><font color="Maroon">R</font> be    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E1:4_1_1"/>
not <font color="Maroon">R</font> is <a href="realset1.html#V1">trivial</a>
 ;<br/>

<b>then consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> )
 <b>by </b><i><a class="ref" href="realset1.html#T14">REALSET1:14</a></i>;<br/>
<a NAME="E4:4_1_1"/><i><font color="Green">E34</font></i>: 
( ( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) &amp; ( <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> or  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">y</font></span><a href="tarski.html#K1">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_1_1"/>
( ( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> ) or ( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) or  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> ) or  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> &amp;  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">y</font></span><a href="tarski.html#K1">}</a></span> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_1_1_1"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="relset_1.html#T26">RELSET_1:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_1_1_2"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="relset_1.html#T26">RELSET_1:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1_3"><b>suppose </b></a><a NAME="E1:4_1_1_1_3"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_1_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="relset_1.html#T27">RELSET_1:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1_4"><b>suppose </b></a><a NAME="E1:4_1_1_1_4"/><i><font color="Green">E35</font></i>: 
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> &amp;  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">y</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_1_1_1_4"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:4_1_1_1_4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:4_1_1_1_4"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><a NAME="E8:4_1_1_1_4"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:4_1_1_1_4"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><a NAME="E11:4_1_1_1_4"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E12:4_1_1_1_4"/>
contradiction
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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