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<b>let </b><font color="Maroon">V</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">V</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:48"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:48"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">V</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D16">CAT_1:def 16</a><br/>

<b>set </b><font color="Maroon">m</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E3:48"/>
 <a href="xboole_0.html#K1">{}</a>  is   <a href="funct_2.html#NM1">Function</a> of  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font>, <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="ens_1.html#E10"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="funct_2.html#T55">FUNCT_2:55</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E4:48"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ens_1.html#E10"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="ens_1.html#E4:48"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E5:48"/>
 <a href="ens_1.html#K7" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="ens_1.html#E10"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="ens_1.html#E4:48"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:48"/><i><font color="Green">E19</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>

<a NAME="E7:48"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">g</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<a NAME="E9:48"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<a NAME="E10:48"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:48"/>
 <a href="ens_1.html#K7" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="ens_1.html#E6:48"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">m'</font> = <font color="Maroon">g</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<a NAME="E13:48"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="ens_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E14:48"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">m</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:48"/><b>then </b>
( <font color="Maroon">m</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">m'</font> <a href="mcart_1.html#K2">`2</a>  is   <a href="funct_2.html#NM1">Function</a> of  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font>, <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="txt" href="ens_1.html#E10:48"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E16:48"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E10"><i><font color="Green">Lemma34</font></i></a>, , <a class="ref" href="partfun1.html#T57">PARTFUN1:57</a></i>;<br/>


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