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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>assume </b><a NAME="E1:25_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="cat_5.html#V3" target="_self">Categorial</a>
 ;<br/>

<a NAME="E2:25_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="cat_5.html#V2" target="_self">categorial</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="cat_1.html#NM4">Category</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> st <font color="Olive">b<sub>1</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="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> ) )
 
<b>by </b><i><a class="ref" href="cat_5.html#D6" target="_self">Def6</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D1" target="_self">CAT_5:def 1</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>4</sub></font> =  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="cat_1.html#NM4">Category</a> <b>by </b><i><a class="txt" href="cat_5.html#E2:25_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="cat_5.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:25_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> st <font color="Maroon">c<sub>2</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="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:25_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="cat_5.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E5:25_1_1"/>
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#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="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>


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