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

<b>let </b><font color="Maroon">C</font> be   <a href="altcat_1.html#NM4">category</a>;<br/>

<a NAME="E1:69"/><i><font color="Green">E24</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="altcat_4.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E2:69"/><i><font color="Green">E25</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="altcat_4.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:69"/><i><font color="Green">E26</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="altcat_2.html#R2">cc=</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="altcat_4.html#T4" target="_self">Th4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:69_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">o1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">o2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:69_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">o1</font>, <font color="Maroon">o2</font> = <font color="Maroon">o2</font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span> <b>by </b><i>, <a class="ref" href="altcat_4.html#T4" target="_self">Th4</a></i>;<br/><b>thus </b><a NAME="E5:69_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:69_1_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , , <a class="ref" href="altcat_2.html#D2">ALTCAT_2:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

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

<b>assume </b><a NAME="E2:69_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E3:69_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E3:69_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E4:69_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
;<br/>
<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font> <b>by </b><i><a class="txt" href="altcat_4.html#E4:69_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E6:69_1_1"/>
<font color="Maroon">n1</font> is <a href="altcat_3.html#V5">epi</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:69_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
<b>by </b><i>, <a class="ref" href="altcat_4.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E8:69_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E3:69_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:69"/><b>then </b>
the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>hence </b><a NAME="E6:69"/>
 <a href="altcat_4.html#K2" target="_self">AllEpi</a> <span class="p1">(<span class="default"><a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="altcat_4.html#K2" target="_self">AllEpi</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="altcat_2.html#T26">ALTCAT_2:26</a></i>;<br/>


</div>
