<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="cat_5.html#V2" target="_self">categorial</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>5</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>;<br/>
<a NAME="E2:43"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  )
 
<b>by </b><i><a class="ref" href="cat_2.html#D2">CAT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<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>7</sub></font>,<font color="Maroon">c<sub>8</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>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>;<br/><b>assume </b><a NAME="E1:43_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E2:43_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp;  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="cat_2.html#D2">CAT_2:def 2</a></i>;<br/><b>hence </b><a NAME="E3:43_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:43_2"/>
(  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="cat_2.html#D2">CAT_2:def 2</a></i>;<br/><b>hence </b><a NAME="E2:43_2"/>
 <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="cat_1.html#V1">strict</a> <a href="cat_5.html#V3" target="_self">Categorial</a> <a href="cat_1.html#NM4">Category</a><b> such that </b><br/><a NAME="E7:43"/><i><font color="Green">E28</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds <br/>( <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> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>7</sub></font> iff <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="cat_5.html#E4:43"><i><font color="Green">E27</font></i></a>, <a class="ref" href="cat_5.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E8:43"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="cat_5.html#V4" target="_self">full</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="cat_1.html#NM4">Category</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D8" target="_self">CAT_5:def 8</a><br/>

<b>assume </b><a NAME="E1:43_3"/>
( <font color="Maroon">c<sub>8</sub></font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E7:43"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>;<br/>

<a NAME="E3:43_3"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp;  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  )
 
<b>by </b><i><a class="ref" href="cat_2.html#D2">CAT_2:def 2</a></i>;<br/>
<a NAME="E4:43_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:43_3"/><b>then </b>
<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>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E7:43"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:43_3"/>
<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>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:43"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="cat_1.html#V1">strict</a> <a href="cat_5.html#V3" target="_self">Categorial</a> <a href="cat_5.html#V4" target="_self">full</a> <a href="cat_1.html#NM4">Category</a> st the <a href="cat_1.html#U1">Objects</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cat_5.html#E7:43"><i><font color="Green">E28</font></i></a></i>;<br/>


</div>
