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<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:48"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:48"/><b>then </b><i><font color="Green" title="E34">A1</font></i>: 
 <a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<b>let </b><font color="Maroon" title="c3">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D16" title="CAT_1:def.16">CAT_1:def 16</a><br/>

<b>set </b><font color="Maroon" title="c4">m</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E3:48"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font>, <a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="ens_1.html#E2:48"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="funct_2.html#T55" title="FUNCT_2:th.55">FUNCT_2:55</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<a NAME="E4:48"/><b>then </b><i><font color="Green" title="E35">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ens_1.html#E2:48"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="ens_1.html#T15" target="_self" title="ENS_1:th.15">Th15</a></i>;<br/>
<a NAME="E5:48"/>
 <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="ens_1.html#E2:48"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="ens_1.html#T15" target="_self" title="ENS_1:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E6:48"/><i><font color="Green" title="E36">A3</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="ens_1.html#T27" target="_self" title="ENS_1:th.27">Th27</a></i>;<br/>

<a NAME="E7:48"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="ens_1.html#E4:48"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="ens_1.html#T27" target="_self" title="ENS_1:th.27">Th27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">f</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <b>by </b><i><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">f</font>
;<br/>

<b>let </b><font color="Maroon" title="c6">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>

<a NAME="E9:48"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="ens_1.html#E6:48"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></i>;<br/>
<a NAME="E10:48"/><b>then </b><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T27" target="_self" title="ENS_1:th.27">Th27</a></i>;<br/>
<a NAME="E11:48"/>
 <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="ens_1.html#T17" target="_self" title="ENS_1:th.17">Th17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">m</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c8">m'</font> = <font color="Maroon" title="c6">g</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="ens_1.html#E4:48"><i><font color="Green" title="E35">A2</font></i></a></i>;<br/>
<a NAME="E13:48"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c8">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">m'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="ens_1.html#E10:48"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="ens_1.html#T16" target="_self" title="ENS_1:th.16">Th16</a></i>;<br/>
<a NAME="E14:48"/>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c7">m</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c7">m</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T8" target="_self" title="ENS_1:th.8">Th8</a></i>;<br/>
<a NAME="E15:48"/><b>then </b>
( <font color="Maroon" title="c7">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c8">m'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font>, <a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E10:48"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="ens_1.html#E13:48"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="ens_1.html#E29"><i><font color="Green" title="E20">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E16:48"/>
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E2:48"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="ens_1.html#E13:48"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="partfun1.html#T57" title="PARTFUN1:th.57">PARTFUN1:57</a></i>;<br/>


</div>
