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<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>set </b><font color="Maroon">A</font> =  <a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font>;<br/>
<b>thus </b><a NAME="E1:9"/>
the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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>consider </b><font color="Maroon">o'</font> being   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>
<a NAME="E2:9_1"/><i><font color="Green">E7</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<b>assume </b><a NAME="E3:9_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>
 ;<br/>

<b>then consider </b><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><br/><a NAME="E5:9_1"/><i><font color="Green">E25</font></i>: 
<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">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E6:9_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:9_1"><i><font color="Green">E8</font></i></a>, , <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E7:9_1"/>
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <span class="p2">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>,the <a href="cat_1.html#U2">Morphisms</a> of <span class="p2">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">x11</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x12</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:9_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x11</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">x12</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> )
 <b>and </b><br/><a NAME="E10:9_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x11</font>,<font color="Maroon">x12</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E11:9_1"/>
( <font color="Maroon">x11</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">o'</font> &amp; <font color="Maroon">x12</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">o'</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:9_1"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o'</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o'</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:9_1"><i><font color="Green">E8</font></i></a>, , , , <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<a NAME="E13:9_1"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o'</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E2:9_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="cat_1.html#T41">CAT_1:41</a></i>;<br/>
<a NAME="E14:9_1"/><b>then </b>
( <font color="Maroon">x11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">x12</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span></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/>


</div><b>end;</b></div>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">m</font> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>
<a NAME="E4:9"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<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>assume </b><a NAME="E5:9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E6:9"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div>
