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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</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="E1:21"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:21"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:21"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 <b>and </b><br/><a NAME="E5:21"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:21"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="mcart_1.html#K2">`2</a>  <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>3</sub></font> )
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:21"><i><font color="Green">E8</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="mcart_1.html#K2">`2</a>  as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="altcat_2.html#E4:21"><i><font color="Green">E8</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E8:21"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:21"><i><font color="Green">E8</font></i></a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E9:21"/>
( <font color="Maroon">c<sub>7</sub></font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="altcat_2.html#E6:21"><i><font color="Green">E10</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<a NAME="E10:21"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K4">*</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>4</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E6:21"><i><font color="Green">E10</font></i></a>, <a class="ref" href="cat_1.html#T51">CAT_1:51</a></i>;<br/>
<b>hence </b><a NAME="E11:21"/>
<font color="Maroon">c<sub>5</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>4</sub></font>
 <b>by </b><i><a class="txt" href="altcat_2.html#E3:21"><i><font color="Green">E7</font></i></a>, <a class="txt" href="altcat_2.html#E5:21"><i><font color="Green">E9</font></i></a>, <a class="txt" href="altcat_2.html#E8:21"><i><font color="Green">E11</font></i></a>, <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>;<br/>


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