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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#NM1">Object</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>given </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:51"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:51"/><i><font color="Green">E36</font></i>: 
 <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:51"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D15">CAT_1:def 15</a><br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font>, <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</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">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E4:51"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</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">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</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">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="ens_1.html#E2:51"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ens_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E5:51"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</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">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="ens_1.html#T27" target="_self">Th27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</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">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>thus </b><a NAME="E7:51"/><i><font color="Green">E38</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <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#E4:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="ens_1.html#T27" target="_self">Th27</a></i>;<br/>

<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E8:51"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="ens_1.html#E7:51"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<a NAME="E9:51"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>8</sub></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">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E10:51"/>
 <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">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ens_1.html#T17" target="_self">Th17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</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">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="ens_1.html#E4:51"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E12:51"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>10</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#K14" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></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="txt" href="ens_1.html#E9:51"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ens_1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E13:51"/>
<font color="Maroon">c<sub>9</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> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></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#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E14:51"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>4</sub></font>, <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E9:51"><i><font color="Green">E39</font></i></a>, <a class="txt" href="ens_1.html#E12:51"><i><font color="Green">E40</font></i></a>, <a class="txt" href="ens_1.html#E29"><i><font color="Green">Lemma20</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E15:51"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="ens_1.html#E2:51"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ens_1.html#E12:51"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_2.html#T66">FUNCT_2:66</a></i>;<br/>


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