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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">C</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">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font>;<br/>





<b>thus </b><a NAME="E1:73"/>
( <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1"/><i><font color="Green" title="E44">A1</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:73_1"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 
;<br/>
<a NAME="E3:73_1"/><b>then </b>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T77" target="_self" title="RFUNCT_1:th.77">Th77</a></i>;<br/>
<a NAME="E4:73_1"/><b>then </b><i><font color="Green" title="E45">A2</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">c</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> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:73_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#T48" title="FUNCT_3:th.48">FUNCT_3:48</a></i>;<br/>
<b>hence </b><a NAME="E6:73_1"/>
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="rfunct_1.html#E1:73_1"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="rfunct_1.html#E4:73_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:73"/><i><font color="Green" title="E44">A3</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>and </b><br/><a NAME="E3:73"/><i><font color="Green" title="E45">A4</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>thus </b><a NAME="E4:73"/>
contradiction
 <b>by </b><i><a class="txt" href="rfunct_1.html#E2:73"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="rfunct_1.html#E3:73"><i><font color="Green" title="E45">A4</font></i></a></i>;<br/>


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