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<div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:33_1_1"/><i><font color="Green">E20</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> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="cat_1.html#NM4">Category</a><br/>for <font color="Olive">b<sub>6</sub></font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font><br/>for <font color="Olive">b<sub>7</sub></font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</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>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">b<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">b<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="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">b<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <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>3</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>7</sub></font> <a href="cat_1.html#K14">*</a> <font color="Olive">b<sub>6</sub></font></span>)</span></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>thus </b><a NAME="E2:33_1_1"/>
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="cat_5.html#V2" target="_self">categorial</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D6" target="_self">CAT_5:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D5" target="_self">CAT_5:def 5</a><br/>

<a NAME="E1:33_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="cat_2.html#T12">CAT_2:12</a></i>;<br/>
<b>hence </b><a NAME="E2:33_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="cat_1.html#NM4">Category</a>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:33_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="cat_5.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="cat_2.html#T12">CAT_2:12</a></i>;<br/><b>assume </b><a NAME="E2:33_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E3:33_1_1_2"/><b>then </b>
<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>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="cat_5.html#D6" target="_self">Def6</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_2"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</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>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="cat_2.html#T14">CAT_2:14</a></i>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>assume </b><a NAME="E2:33_1_1_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E3:33_1_1_3"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="cat_2.html#T15">CAT_2:15</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> st <font color="Maroon">c<sub>3</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>5</sub></font>,<font color="Maroon">c<sub>6</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="ref" href="cat_5.html#D6" target="_self">Def6</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="cat_2.html#T14">CAT_2:14</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>



<b>assume </b><a NAME="E6:33_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>3</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>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>4</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>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<a NAME="E7:33_1_1"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K17">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K18">`12</a>  )
 
<b>by </b><i><a class="ref" href="cat_5.html#T13" target="_self">Th13</a>, <a class="ref" href="cat_2.html#T15">CAT_2:15</a></i>;<br/>
<a NAME="E8:33_1_1"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E6:33_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="cat_2.html#T17">CAT_2:17</a></i>
<br/>.= 
<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>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E6:33_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="cat_5.html#D6" target="_self">Def6</a></i>
;<br/><br/>


</div>
