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<b>let </b><font color="Maroon">h1</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font>, <font color="Maroon">h2</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:37_3_2"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">m2</font>, <font color="Olive">m1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">m2</font>,<font color="Olive">m1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font> iff  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Olive">m2</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Olive">m1</font> )
 <b>and </b><br/><a NAME="E2:37_3_2"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">m2</font>, <font color="Olive">m1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font>  st  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Olive">m2</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Olive">m1</font> holds <br/><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">m2</font>,<font color="Olive">m1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">m2</font> <a href="ens_1.html#K6" target="_self">*</a> <font color="Olive">m1</font>
 <b>and </b><br/><a NAME="E3:37_3_2"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">m2</font>, <font color="Olive">m1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">m2</font>,<font color="Olive">m1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font> iff  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Olive">m2</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Olive">m1</font> )
 <b>and </b><br/><a NAME="E4:37_3_2"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">m2</font>, <font color="Olive">m1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font>  st  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Olive">m2</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Olive">m1</font> holds <br/><font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">m2</font>,<font color="Olive">m1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">m2</font> <a href="ens_1.html#K6" target="_self">*</a> <font color="Olive">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">E81</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>thus </b><a NAME="E2:37_3_2_1"/><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><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> ;<br/><b>thus </b><a NAME="E3:37_3_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">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">E83</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">m2</font> = <font color="Maroon">x</font>, <font color="Maroon">m1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:37_3_2_1_1"/>
 <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">m1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:37_3_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E4:37_3_2_1"/><i><font color="Green">E84</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">m2</font> = <font color="Maroon">x</font>, <font color="Maroon">m1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:37_3_2_1"/>
 <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">m1</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:37_3_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E6:37_3_2"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T110">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">m</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><b>assume </b><a NAME="E1:37_3_2_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font>
 ;<br/><b>consider </b><font color="Maroon">m2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font>, <font color="Maroon">m1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:37_3_2_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m2</font>,<font color="Maroon">m1</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><a NAME="E4:37_3_2_2"/>
 <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">m1</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:37_3_2_2"/><b>then </b>
( <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m2</font>,<font color="Maroon">m1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font> <a href="ens_1.html#K6" target="_self">*</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m2</font>,<font color="Maroon">m1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font> <a href="ens_1.html#K6" target="_self">*</a> <font color="Maroon">m1</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:37_3_2_2"/>
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:37_3_2"/>
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


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