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<div class="add">

<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>assume </b><b>that </b><br/><a NAME="E1:52"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E2:52"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="cat_1.html#V6">terminal</a>
 ;<br/>

<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E39</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>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:52_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:52_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</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#E1:52"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#T41">ZFMISC_1:41</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="ens_1.html#E2:52_1_1"><i><font color="Green">E40</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="ens_1.html#K13" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E5:52_1_1"/>
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K13" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<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>  &amp;  <a href="ens_1.html#K13" target="_self">@</a> <font color="Maroon">c<sub>4</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#E2:52"><i><font color="Green">E37</font></i></a>, <a class="txt" href="ens_1.html#E3:52_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="cat_1.html#D15">CAT_1:def 15</a></i>;<br/><a NAME="E6:52_1_1"/><b>then </b>
(  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K13" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="ens_1.html#K14" target="_self">@</a> <span class="p1">(<span class="default"><a href="ens_1.html#K13" target="_self">@</a> <font color="Maroon">c<sub>4</sub></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#E51"><i><font color="Green">Lemma31</font></i></a></i>;<br/><b>hence </b><a NAME="E7:52_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ens_1.html#E1:52_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#T14">FUNCT_2:14</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:52_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="E2:52_2"/><b>then </b>
 <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:52_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:52_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ens_1.html#E3:52"><i><font color="Green">E38</font></i></a>, <a class="ref" href="zfmisc_1.html#T41">ZFMISC_1:41</a></i>;<br/><a NAME="E6:52_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <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#E4:52_2"><i><font color="Green">E39</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <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="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</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="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/><a NAME="E8:52_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <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="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</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#E4:52_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/><a NAME="E10:52_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E11:52_2"/>
(  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><a NAME="E12:52_2"/><b>then </b><i><font color="Green">E43</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>2</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>2</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> &amp; <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>2</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>6</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>2</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#T15" target="_self">Th15</a></i>;<br/><a NAME="E13:52_2"/>
 <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>2</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>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>2</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>8</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>2</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>6</sub></font></span><a href="domain_1.html#K1">]</a></span> 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#E12:52_2"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E15:52_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <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>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="ens_1.html#E12:52_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="ens_1.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E16:52_2"/><b>then </b>
(  <a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="ens_1.html#K15" target="_self">@</a> <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>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> =  <a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">c<sub>8</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>2</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>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E19:52_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ens_1.html#E2:52"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cat_1.html#D15">CAT_1:def 15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1"><b>now </b></a><div class="add"><b>thus </b><font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="ens_1.html#E19:52_2"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="ens_1.html#E19:52_2"><i><font color="Green">E44</font></i></a></i>
;<br/><b>thus </b><a NAME="E2:52_2_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <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>2</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> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <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>2</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>6</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 ;<br/><a NAME="E3:52_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="ens_1.html#E4:52_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="ens_1.html#E8:52_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="ens_1.html#E10:52_2"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E4:52_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="ens_1.html#E5:52_2"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E21:52_2"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:52"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <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="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


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