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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>











<b>assume </b><a NAME="E1:3"/>
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 <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/>

<a NAME="E2:3"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</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="ref" href="commacat.html#D5" target="_self">Def5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:3"/><i><font color="Green">E4</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"><span class="p2"><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>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>9</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="Maroon">c<sub>7</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="Maroon">c<sub>8</sub></font></span>)</span> )
 ;<br/>
<a NAME="E5:3"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K14">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K15">`12</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="commacat.html#E4:3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="commacat.html#E6"><i><font color="Green">Lemma1</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E6:3"/>
( <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"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K14">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K15">`12</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K3">`2</a>  <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K14">`11</a> </span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K15">`12</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="commacat.html#E4:3"><i><font color="Green">E4</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>


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