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<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:87_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2">disjoin</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Olive">x</font>] ) )
 ;<br/>

<b>reconsider </b><font color="Maroon">fa</font> = <font color="Maroon">a</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> ;<br/>
<b>take </b>
<font color="Maroon">fa</font>
;<br/>

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

<b>take </b><font color="Maroon">g</font> =  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">fa</font>;<br/>

<b>thus </b><a NAME="E3:87_2"/>
( <font color="Maroon">fa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">fa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">fa</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>  )
 ;<br/>

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

<b>assume </b><a NAME="E4:87_2"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">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> 
 ;<br/>

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

<a NAME="E5:87_2"/><i><font color="Green">E56</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">fa</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E6:87_2"/><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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="mcart_1.html#K20">`22</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">fa</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>;<br/>
<a NAME="E7:87_2"/><i><font color="Green">E62</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="hidden.html#R1">=</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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:87_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">fa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">fa</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="E9:87_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2">disjoin</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T33">CARD_3:33</a></i>;<br/>
<a NAME="E10:87_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">fa</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">fa</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<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="Maroon">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">fa</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="altcat_1.html#T24">ALTCAT_1:24</a></i>
;<br/><br/>


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