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

<b>let </b><font color="Maroon">V</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">V</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:55"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">f</font> is <a href="cat_1.html#V4">epi</a>
 ;<br/>

<b>given </b><font color="Maroon">B</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:55"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>and </b><br/><a NAME="E3:55"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E4:55"/><i><font color="Green">E20</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:55"/>
 <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>set </b><font color="Maroon">A</font> =  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:55_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E2:55_1"/><i><font color="Green">E63</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">g1</font> = <span class="p1">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x1</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">B</font> <b>by </b><i>, <a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<b>set </b><font color="Maroon">h</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x2</font>;<br/>
<b>set </b><font color="Maroon">g2</font> = <font color="Maroon">g1</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x2</font></span>)</span>;<br/>
<a NAME="E4:55_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:55_1"/><i><font color="Green">E75</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>then </b><i><font color="Green">E76</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
 <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="ens_1.html#T8" target="_self">Th8</a>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/>
<a NAME="E7:55_1"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="ens_1.html#T8" target="_self">Th8</a>, <a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E8:55_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E9:55_1"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="enumset1.html#T41">ENUMSET1:41</a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E10:55_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g2</font> = <font color="Maroon">g1</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x2</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">B</font> <b>by </b><i>, <a class="ref" href="ens_1.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m1</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">B</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">g1</font></span><a href="domain_1.html#K1">]</a></span>, <font color="Maroon">m2</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">B</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">g2</font></span><a href="domain_1.html#K1">]</a></span> 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>, </i>;<br/>
<b>set </b><font color="Maroon">f1</font> =  <a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font>;<br/>
<b>set </b><font color="Maroon">f2</font> =  <a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font>;<br/>
<a NAME="E13:55_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">m1</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">m2</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">m2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">m2</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E14:55_1"/><b>then </b>
(  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp;  <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> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E15:55_1"/><b>then </b><i><font color="Green">E78</font></i>: 
(  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 
;<br/>
<a NAME="E16:55_1"/><b>then </b><i><font color="Green">E79</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E17:55_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font> <a href="mcart_1.html#K2">`2</a>  )
 
<b>by </b><i><a class="ref" href="ens_1.html#T11" target="_self">Th11</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>set </b><font color="Maroon">h1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">h2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">f1f</font> = <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">f2f</font> = <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2"><b>now </b></a><div class="add"><a NAME="E1:55_1_2"/><i><font color="Green">E81</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> &amp;  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> &amp;  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="ens_1.html#D7" target="_self">Def7</a>, </i>;<br/><b>hence </b><a NAME="E2:55_1_2"/>
( <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:55_1_2_1"/><i><font color="Green">E82</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ens_1.html#T14" target="_self">Th14</a>, </i>;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>set </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>;<br/><b>assume </b><a NAME="E2:55_1_2_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E3:55_1_2_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:55_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:55_1_2_1"/><b>then </b>
( <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="txt" href="ens_1.html#E15"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="ens_1.html#T13" target="_self">Th13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:55_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 <b>by </b><i>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><b>hence </b><a NAME="E7:55_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E17"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:55_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:55_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 
<b>by </b><i><a class="ref" href="ens_1.html#T9" target="_self">Th9</a>, <a class="ref" href="ens_1.html#D7" target="_self">Def7</a>, <a class="ref" href="ens_1.html#T12" target="_self">Th12</a>, <a class="ref" href="ens_1.html#T5" target="_self">Th5</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E20:55_1"/><b>then </b>
(  <a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m2</font> &amp; <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 
<b>by </b><i>, <a class="ref" href="ens_1.html#T8" target="_self">Th8</a>, <a class="ref" href="ens_1.html#T9" target="_self">Th9</a>, <a class="txt" href="ens_1.html#E17"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="ens_1.html#T12" target="_self">Th12</a>, <a class="ref" href="cat_1.html#D11">CAT_1:def 11</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/>
<b>hence </b><a NAME="E21:55_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:55"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ens_1.html#D9" target="_self">ENS_1:def 9</a><br/>


</div>
