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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:56"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ens_1.html#D9" target="_self">ENS_1:def 9</a><br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D11">CAT_1:def 11</a><br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:56"/><i><font color="Green">E38</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E3:56"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E4:56"/><i><font color="Green">E40</font></i>: 
(  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="ens_1.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E5:56"/><b>then </b>
( <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E2:56"><i><font color="Green">E38</font></i></a>, <a class="ref" href="ens_1.html#D7" target="_self">Def7</a>, <a class="ref" href="ens_1.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E6:56"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E3:56"><i><font color="Green">E39</font></i></a>, <a class="txt" href="ens_1.html#E4:56"><i><font color="Green">E40</font></i></a>, <a class="txt" href="ens_1.html#E17"><i><font color="Green">Lemma10</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E7:56"/><b>then </b>
( <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  &amp;  <a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></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> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></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> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="txt" href="ens_1.html#E2:56"><i><font color="Green">E38</font></i></a>, <a class="txt" href="ens_1.html#E4:56"><i><font color="Green">E40</font></i></a>, <a class="ref" href="ens_1.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T156">FUNCT_1:156</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>4</sub></font> = 
 <a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="ens_1.html#E2:56"><i><font color="Green">E38</font></i></a>, <a class="txt" href="ens_1.html#E4:56"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>

;<br/><br/>


</div>
