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

<b>set </b><font color="Maroon">B</font> =  <a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">FF</font> =  <a href="yellow18.html#K5" target="_self">Concretization</a> <font color="Maroon">A</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:94_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> holds  <span class="p1">(<span class="default"><a href="yellow18.html#K5" target="_self">Concretization</a> <font color="Maroon">A</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>)
 
<b>by </b><i/>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>,  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>,  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K5" target="_self">Concretization</a> <font color="Maroon">A</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K5" target="_self">Concretization</a> <font color="Maroon">A</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> = <span class="p1">(<span class="default"><a href="yellow18.html#K5" target="_self">Concretization</a> <font color="Maroon">A</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">a<sub>3</sub></font>;<br/>
<a NAME="E2:94_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font> holds  <span class="p1">(<span class="default"><a href="yellow18.html#K5" target="_self">Concretization</a> <font color="Maroon">A</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>)
 
;<br/>
<a NAME="E3:94_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>  st <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>) holds <br/><font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 
;<br/>
<a NAME="E4:94_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font>  st <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">g</font>) holds <br/><font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:94_1_1_1"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>, <font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<a NAME="E2:94_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K5" target="_self">Concretization</a> <font color="Maroon">A</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K5" target="_self">Concretization</a> <font color="Maroon">A</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:94_1_1_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">g</font>) implies <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:94_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font></span>)</span>  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font>  ex <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> ex <font color="Olive">g</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">c</font>,<font color="Olive">d</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">c</font>) &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d</font>) &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">c</font>,<font color="Olive">d</font>,<font color="Olive">g</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:94_1_1_2"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">a</font>, <font color="Maroon">d</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>consider </b><font color="Maroon">fa</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">fb</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">g</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">fa</font>,<font color="Maroon">fb</font><b> such that </b><br/><a NAME="E4:94_1_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">fa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">fb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">fa</font>,<font color="Maroon">fb</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>and </b><br/><a NAME="E5:94_1_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">o</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o</font>,<font color="Maroon">fa</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">h</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o</font>,<font color="Maroon">fa</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">h</font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,<font color="Maroon">fa</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Olive">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,<font color="Maroon">fb</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">d</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<a NAME="E7:94_1_1_2"/>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">c</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K5" target="_self">Concretization</a> <font color="Maroon">A</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c</font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, , <a class="ref" href="altcat_1.html#D19">ALTCAT_1:def 19</a></i>;<br/>
<a NAME="E8:94_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K5" target="_self">Concretization</a> <font color="Maroon">A</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c</font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c</font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="txt" href="yellow18.html#E5:94_1_1_2"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E9:94_1_1_2"/>
( <span class="p1">(<span class="default"><a href="yellow18.html#K5" target="_self">Concretization</a> <font color="Maroon">A</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="yellow18.html#K5" target="_self">Concretization</a> <font color="Maroon">A</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:94_1_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c</font>) &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">d</font>) &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">g</font>) )
 <b>by </b><i>, , <a class="txt" href="yellow18.html#E8:94_1_1_2"><i><font color="Green">E54</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:94_1_1"/>
 <a href="yellow18.html#K5" target="_self">Concretization</a> <font color="Maroon">A</font> is <a href="functor0.html#V21">bijective</a>
 <b>from </b><i><a class="ref" href="yellow18.html#S10" target="_self">YELLOW18:sch 10</a>(, , <a class="txt" href="yellow18.html#E5:94_1_1"><i><font color="Green">E48</font></i></a>, , );</i><br/>


</div>
