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<a NAME="E1:2_1_1"/><i><font color="Green">E3</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"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="commacat.html#E1:2"><i><font color="Green">E2</font></i></a></i>;<br/>
<a NAME="E2:2_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> </span>}</span> 
 ;<br/>

<a NAME="E2:2_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> st <br/>( <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"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:2_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:2_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="commacat.html#E1:2_1_1"><i><font color="Green">E3</font></i></a></i>;<br/>


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