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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:74"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<i><font color="Green">E30</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">C</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a></i>
<br/>.= 
<font color="Maroon">C</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E1:74_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_2_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_1_1"><b>suppose </b></a><a NAME="E1:74_2_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:74_2_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E3:74_2_1_1_1"/>
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T17" target="_self">Th17</a>, <a class="ref" href="rfunct_1.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:74_2_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E5:74_2_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:74_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T25" target="_self">Th25</a>, <a class="ref" href="rfunct_1.html#T26" target="_self">Th26</a>, <a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_1_2"><b>suppose </b></a><a NAME="E1:74_2_1_1_2"/><i><font color="Green">E64</font></i>: 
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:74_2_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_2_1_1_2_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or  not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:74_2_1_1_2_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><a NAME="E2:74_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E3:74_2_1_1_2_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T26" target="_self">Th26</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:74_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T25" target="_self">Th25</a>, <a class="ref" href="rfunct_1.html#T29" target="_self">Th29</a>, <a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:74_2_1_1_2_1_1_2"/><i><font color="Green">E72</font></i>: 
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><a NAME="E2:74_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E3:74_2_1_1_2_1_1_2"/>
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T24" target="_self">Th24</a>, <a class="ref" href="rfunct_1.html#T26" target="_self">Th26</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:74_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T25" target="_self">Th25</a>, <a class="ref" href="rfunct_1.html#T29" target="_self">Th29</a>, <a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:74_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:74_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T22" target="_self">Th22</a>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:74"/>
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" target="_self">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rfunct_1.html#K5" target="_self">chi</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T21" target="_self">Th21</a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


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