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

<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> , <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> ;<br/>

<b>set </b><font color="Maroon">f</font> = <font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">Y</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font>;<br/>
<a NAME="E1:9"/><i><font color="Green">E24</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>thus </b><a NAME="E2:9"/>
(  not <font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">Y</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:9_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 <b>and </b><br/><a NAME="E2:9_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a><br/>

<b>assume </b><a NAME="E3:9_1"/>
<font color="Maroon">X</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:9_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E6:9_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:9_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


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

<b>assume </b><a NAME="E3:9"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 ;<br/>

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a><br/>

<b>assume </b><a NAME="E4:9"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E5:9"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:9"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E7:9"/>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 ;<br/>


</div>
