<?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">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>, <font color="Maroon">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:104"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>and </b><br/><a NAME="E2:104"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#R1" target="_self">tolerates</a> <font color="Maroon">g</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_1_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Y</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="104_1_1_1"><b>suppose </b></a><a NAME="E1:104_1_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:104_1_1_1"/><b>then </b>
( <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> is <a href="partfun1.html#V1" target="_self">total</a> &amp; <font color="Maroon">f</font> <a href="partfun1.html#R1" target="_self">tolerates</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> &amp; <font color="Maroon">g</font> <a href="partfun1.html#R1" target="_self">tolerates</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> )
 <b>by </b><i><a class="txt" href="partfun1.html#E13"><i><font color="Green">Lemma49</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E3:104_1_1_1"/>
 ex <font color="Olive">h</font> being  <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> st <br/>( <font color="Olive">h</font> is <a href="partfun1.html#V1" target="_self">total</a> &amp; <font color="Maroon">f</font> <a href="partfun1.html#R1" target="_self">tolerates</a> <font color="Olive">h</font> &amp; <font color="Maroon">g</font> <a href="partfun1.html#R1" target="_self">tolerates</a> <font color="Olive">h</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1_2"><b>suppose </b></a><a NAME="E1:104_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) );<br/><a NAME="E3:104_1_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">y'</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y'</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:104_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_1_1_2_1_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:104_1_1_2_1_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">y'</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E2:104_1_1_2_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> implies <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 ;<br/><a NAME="E3:104_1_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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>
 <b>by </b><i><a class="ref" href="partfun1.html#T31" target="_self">Th31</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:104_1_1_2_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="txt" href="partfun1.html#E6:104_1_1_2"><i><font color="Green">E38</font></i></a></i>;<br/><b>thus </b><a NAME="E5:104_1_1_2_1_1_1_1"/>
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:104_1_1_2_1_1_1_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">y'</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E2:104_1_1_2_1_1_1_2"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> implies <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> ) &amp; ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 <b>by </b><i><a class="ref" href="partfun1.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:104_1_1_2_1_1_1_3"/><i><font color="Green">E36</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">y'</font> = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E2:104_1_1_2_1_1_1_3"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> implies <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> ) &amp; ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 <b>by </b><i><a class="ref" href="partfun1.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1_2_1_1_1_4"><b>suppose </b></a><a NAME="E1:104_1_1_2_1_1_1_4"/>
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:104_1_1_2_1_1_1_4"/>
 ex <font color="Olive">y'</font> being   <a href="hidden.html#M1">set</a>  st <br/>( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> implies <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> ) &amp; ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:104_1_1_2_1"/>
 ex <font color="Olive">y'</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y'</font>]
 ;<br/>


</div><b>end;</b></div><a NAME="E4:104_1_1_2"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 ;<br/><b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:104_1_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E7:104_1_1_2"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="ref" href="partfun1.html#T31" target="_self">Th31</a>, <a class="ref" href="partfun1.html#T30" target="_self">Th30</a>);<br/></i><a NAME="E8:104_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#R1" target="_self">tolerates</a> <font color="Maroon">h</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:104_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font></span>)</span>
 ;<br/>

<a NAME="E2:104_1_1_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</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="E3:104_1_1_2_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E9:104_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#R1" target="_self">tolerates</a> <font color="Maroon">h</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_1_2_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:104_1_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font></span>)</span>
 ;<br/>

<a NAME="E2:104_1_1_2_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</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="E3:104_1_1_2_3"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E10:104_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_1_2_4"><b>proof </b></a><div class="add">

<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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:104_1_1_2_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:104_1_1_2_4"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E4:104_1_1_2_4"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_1_1_2_4_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1_2_4_1_1"><b>suppose </b></a><a NAME="E1:104_1_1_2_4_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 ;<br/><div class="add"><a NAME="E2:104_1_1_2_4_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="partfun1.html#E9:104_1_1_2"><i><font color="Green">E59</font></i></a></i>;<br/><b>hence </b><a NAME="E3:104_1_1_2_4_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="partfun1.html#E3:104_1_1_2_4"><i><font color="Green">E60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1_2_4_1_2"><b>suppose </b></a><a NAME="E1:104_1_1_2_4_1_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 ;<br/><div class="add"><a NAME="E2:104_1_1_2_4_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="partfun1.html#E9:104_1_1_2"><i><font color="Green">E59</font></i></a></i>;<br/><b>hence </b><a NAME="E3:104_1_1_2_4_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="partfun1.html#E3:104_1_1_2_4"><i><font color="Green">E60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1_2_4_1_3"><b>suppose </b></a><a NAME="E1:104_1_1_2_4_1_3"/><i><font color="Green">E66</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 ;<br/><div class="add"><a NAME="E2:104_1_1_2_4_1_3"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="partfun1.html#E9:104_1_1_2"><i><font color="Green">E59</font></i></a></i>;<br/><b>hence </b><a NAME="E3:104_1_1_2_4_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="partfun1.html#E3:104_1_1_2_4"><i><font color="Green">E60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1_2_4_1_4"><b>suppose </b></a><a NAME="E1:104_1_1_2_4_1_4"/>
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 ;<br/><div class="add"><a NAME="E2:104_1_1_2_4_1_4"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="partfun1.html#T32" target="_self">Th32</a>, , <a class="txt" href="partfun1.html#E8:104_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="partfun1.html#E9:104_1_1_2"><i><font color="Green">E59</font></i></a></i>;<br/><b>hence </b><a NAME="E3:104_1_1_2_4_1_4"/>
<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="partfun1.html#T27" target="_self">Th27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">h'</font> = <font color="Maroon">h</font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="partfun1.html#T32" target="_self">Th32</a>, </i>;<br/><a NAME="E12:104_1_1_2"/>
<font color="Maroon">h'</font> is <a href="partfun1.html#V1" target="_self">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#T32" target="_self">Th32</a>, <a class="txt" href="partfun1.html#E2:104"><i><font color="Green">E31</font></i></a></i>;<br/><b>hence </b><a NAME="E13:104_1_1_2"/>
 ex <font color="Olive">h</font> being  <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> st <br/>( <font color="Olive">h</font> is <a href="partfun1.html#V1" target="_self">total</a> &amp; <font color="Maroon">f</font> <a href="partfun1.html#R1" target="_self">tolerates</a> <font color="Olive">h</font> &amp; <font color="Maroon">g</font> <a href="partfun1.html#R1" target="_self">tolerates</a> <font color="Olive">h</font> )
 <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="E4:104"/>
 ex <font color="Olive">h</font> being  <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> st <br/>( <font color="Olive">h</font> is <a href="partfun1.html#V1" target="_self">total</a> &amp; <font color="Maroon">f</font> <a href="partfun1.html#R1" target="_self">tolerates</a> <font color="Olive">h</font> &amp; <font color="Maroon">g</font> <a href="partfun1.html#R1" target="_self">tolerates</a> <font color="Olive">h</font> )
 ;<br/>


</div>
