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

<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> be  <a href="cat_5.html#V3" target="_self">Categorial</a> <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>2</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>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><a NAME="E1:45"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E3:45"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="cat_5.html#V2" target="_self">categorial</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</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="E1:45_1"/>
( <font color="Maroon">c<sub>7</sub></font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>7</sub></font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:45_1"/>
<font color="Maroon">c<sub>7</sub></font> is   <a href="cat_1.html#NM4">Category</a>
 <b>by </b><i><a class="ref" href="cat_5.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>3</sub></font> as  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>consider </b><font color="Maroon">c<sub>8</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><b> such that </b><br/><a NAME="E6:45"/><i><font color="Green">E29</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="cat_5.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E7:45"/>
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> &amp; the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E6:45"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cat_5.html#T21" target="_self">Th21</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> ;<br/>
<a NAME="E11:45"/>
(  <a href="cat_2.html#K8">incl</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R1">=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>9</sub></font> )
 
;<br/>
<a NAME="E12:45"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K8">incl</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> ;<br/>
 <a href="cat_5.html#K7" target="_self">Image</a> <font color="Maroon">c<sub>11</sub></font> = 
 <a href="cat_5.html#K7" target="_self">Image</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="cat_5.html#T9" target="_self">Th9</a></i>
<br/>.= 
 <a href="cat_5.html#K7" target="_self">Image</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E1:45"><i><font color="Green">E28</font></i></a>, <a class="ref" href="cat_5.html#T9" target="_self">Th9</a></i>
;<br/>
<b>hence </b><a NAME="E15:45"/>
 <a href="cat_5.html#K7" target="_self">Image</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_5.html#K7" target="_self">Image</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div>
