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

<b>let </b><font color="Maroon">A</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><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="E1:91"/>
<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>set </b><font color="Maroon">B</font> =  <a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font>;<br/>
<b>let </b><font color="Maroon">F1</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">F2</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:91"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">F1</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="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E3:91"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F2</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="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E4:91"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">F1</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">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> <font color="Maroon">F2</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">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>
 ;<br/>

<a NAME="E5:91"/>
( <font color="Maroon">F1</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">F2</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:91"/><b>then </b><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<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">f</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="E8:91"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">fa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">fb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</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="E9:91"/><i><font color="Green">E48</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">F1</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">f</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>consider </b><font color="Maroon">ga</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">gb</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">ga</font>,<font color="Maroon">gb</font><b> such that </b><br/><a NAME="E11:91"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">ga</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">gb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">ga</font>,<font color="Maroon">gb</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="E12:91"/><i><font color="Green">E50</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">ga</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">ga</font> holds  <font color="Maroon">F2</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">ga</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">gb</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">f</font> = <font color="Maroon">f</font>, <font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> <b>by </b><i><a class="txt" href="yellow18.html#E11:91"><i><font color="Green">E49</font></i></a>, </i>;<br/>
<a NAME="E14:91"/>
( <font color="Maroon">F1</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">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"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="altcat_1.html#K5">*</a> <span class="p3">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<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; <font color="Maroon">f</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">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &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">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">F2</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">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"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <span class="p3">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<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><a class="txt" href="yellow18.html#E11:91"><i><font color="Green">E49</font></i></a>, <a class="txt" href="yellow18.html#E12:91"><i><font color="Green">E50</font></i></a>, , , <a class="ref" href="altcat_1.html#D19">ALTCAT_1:def 19</a></i>;<br/>
<a NAME="E15:91"/><b>then </b><i><font color="Green">E53</font></i>: 
<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><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:91_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">A</font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">a</font>
 ;<br/><b>then consider </b><font color="Maroon">bb</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">ff</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">bb</font>,<font color="Maroon">a</font><b> such that </b><br/><a NAME="E3:91_1"/><i><font color="Green">E54</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">bb</font>,<font color="Maroon">a</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">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ff</font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">bb</font>,<font color="Maroon">a</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#E6:91"><i><font color="Green">E42</font></i></a></i>;<br/><b>thus </b><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">ff</font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">bb</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><a class="txt" href="yellow18.html#E11:91"><i><font color="Green">E49</font></i></a>, <a class="txt" href="yellow18.html#E12:91"><i><font color="Green">E50</font></i></a>, <a class="txt" href="yellow18.html#E3:91_1"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="txt" href="yellow18.html#E15:91"><i><font color="Green">E53</font></i></a>, <a class="txt" href="yellow18.html#E3:91_1"><i><font color="Green">E54</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:91"/>
<font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="txt" href="yellow18.html#E9:91"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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