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

<b>let </b><font color="Maroon">A1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">A2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A1</font>,<font color="Maroon">A2</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">B1</font>,<font color="Maroon">B2</font>;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:18"/>
( ( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; ( <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 
;<br/>
<a NAME="E3:18"/><b>then </b><i><font color="Green">E32</font></i>: 
( ( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) ) or (  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T26">RELSET_1:26</a></i>;<br/>
<a NAME="E4:18"/>
( ( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; ( <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 
;<br/>
<a NAME="E5:18"/><b>then </b><i><font color="Green">E33</font></i>: 
( ( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) ) or (  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T26">RELSET_1:26</a></i>;<br/>
<a NAME="E6:18"/><i><font color="Green">E34</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font> )
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E7:18"/><b>then </b><i><font color="Green">E39</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B2</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T14">RELAT_1:14</a>, <a class="ref" href="relat_1.html#T27">RELAT_1:27</a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E8:18"/><b>then </b><i><font color="Green">E51</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:18"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">f</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">g</font> is   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font>,<font color="Maroon">A2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T27">GRFUNC_1:27</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E54</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:18_1_1"/>
<font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:18_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> )
 <b>by </b><i>, , <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:18_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1"/>
<font color="Maroon">A2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E4:18_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:18_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:18_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font>
 ;<br/>

<a NAME="E3:18_1_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> )
 
<b>by </b><i>, , <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:18_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<a NAME="E5:18_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1"/>
( <font color="Maroon">A2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">A2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_1"><b>case </b></a><a NAME="E1:18_2_1_1"/>
<font color="Maroon">A2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font>
 <b>by </b><i>, , , <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2"><b>case </b></a><a NAME="E1:18_2_1_2"/>
<font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font>
 <b>by </b><i><a class="ref" href="yellow20.html#T1" target="_self">Th1</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <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="18_2_1_3"><b>case </b></a><a NAME="E1:18_2_1_3"/>
( <font color="Maroon">A2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_3"/>
<font color="Maroon">f</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:18"/>
<font color="Maroon">f</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B1</font>,<font color="Maroon">A2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B2</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="grfunc_1.html#T27">GRFUNC_1:27</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>


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