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

<b>let </b><font color="Maroon" title="c2">h1</font>, <font color="Maroon" title="c3">h2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:37_3_2"/><i><font color="Green" title="E24">A10</font></i>: 
for <font color="Olive" title="b1">m2</font>, <font color="Olive" title="b2">m1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">m2</font>,<font color="Olive" title="b2">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font> iff  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Olive" title="b1">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Olive" title="b2">m1</font> )
 <b>and </b><br/><a NAME="E2:37_3_2"/><i><font color="Green" title="E25">A11</font></i>: 
for <font color="Olive" title="b1">m2</font>, <font color="Olive" title="b2">m1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>  st  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Olive" title="b1">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Olive" title="b2">m1</font> holds <br/><font color="Maroon" title="c2">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">m2</font>,<font color="Olive" title="b2">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Olive" title="b2">m1</font>
 <b>and </b><br/><a NAME="E3:37_3_2"/><i><font color="Green" title="E26">A12</font></i>: 
for <font color="Olive" title="b1">m2</font>, <font color="Olive" title="b2">m1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">m2</font>,<font color="Olive" title="b2">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font> iff  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Olive" title="b1">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Olive" title="b2">m1</font> )
 <b>and </b><br/><a NAME="E4:37_3_2"/><i><font color="Green" title="E27">A13</font></i>: 
for <font color="Olive" title="b1">m2</font>, <font color="Olive" title="b2">m1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>  st  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Olive" title="b1">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Olive" title="b2">m1</font> holds <br/><font color="Maroon" title="c3">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">m2</font>,<font color="Olive" title="b2">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Olive" title="b2">m1</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:37_3_2_1"/><i><font color="Green" title="E28">A14</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>thus </b><a NAME="E2:37_3_2_1"/><i><font color="Green" title="E29">A15</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E3:37_3_2_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_3_2_1_1"/><i><font color="Green" title="E30">A16</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">m2</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">m1</font> = <font color="Maroon" title="c5">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="ens_1.html#E1:37_3_2_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:37_3_2_1_1"/>
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c7">m1</font>
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:37_3_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="ens_1.html#E1:37_3_2_1_1"><i><font color="Green" title="E30">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:37_3_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E3:37_3_2"><i><font color="Green" title="E26">A12</font></i></a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E4:37_3_2_1"/><i><font color="Green" title="E30">A17</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">m2</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">m1</font> = <font color="Maroon" title="c5">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="ens_1.html#E2:37_3_2_1"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:37_3_2_1"/>
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c7">m1</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E3:37_3_2"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="ens_1.html#E4:37_3_2_1"><i><font color="Green" title="E30">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E7:37_3_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E1:37_3_2"><i><font color="Green" title="E24">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:37_3_2"/><b>then </b><i><font color="Green" title="E28">A18</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T110" title="ZFMISC_1:th.110">ZFMISC_1:110</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/><b>assume </b><a NAME="E1:37_3_2_2"/><i><font color="Green" title="E29">A19</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font>
 ;<br/><b>consider </b><font color="Maroon" title="c5">m2</font>, <font color="Maroon" title="c6">m1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:37_3_2_2"/><i><font color="Green" title="E30">A20</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">m2</font>,<font color="Maroon" title="c6">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/><a NAME="E4:37_3_2_2"/>
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c6">m1</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E1:37_3_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="ens_1.html#E1:37_3_2_2"><i><font color="Green" title="E29">A19</font></i></a>, <a class="txt" href="ens_1.html#E3:37_3_2_2"><i><font color="Green" title="E30">A20</font></i></a></i>;<br/><a NAME="E5:37_3_2_2"/><b>then </b>
( <font color="Maroon" title="c2">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">m2</font>,<font color="Maroon" title="c6">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c6">m1</font> &amp; <font color="Maroon" title="c3">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">m2</font>,<font color="Maroon" title="c6">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c6">m1</font> )
 <b>by </b><i><a class="txt" href="ens_1.html#E2:37_3_2"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="ens_1.html#E4:37_3_2"><i><font color="Green" title="E27">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E6:37_3_2_2"/>
<font color="Maroon" title="c2">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E3:37_3_2_2"><i><font color="Green" title="E30">A20</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:37_3_2"/>
<font color="Maroon" title="c2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h2</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E6:37_3_2"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


</div>
