<?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> be    <a href="cat_2.html#M3" target="_self">Subcategory</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>thus </b><a NAME="E1:28"/>
(  <a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font> is <a href="cat_1.html#V9">full</a> implies for <font color="Olive">b<sub>1</sub></font>, <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><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/> <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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> st  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>(  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ex <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>4</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D23">CAT_1:def 23</a><br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E2:28_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:28_1_1"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="cat_2.html#E2:28_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="cat_2.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E2:28_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 ;<br/><a NAME="E3:28_1_1"/><i><font color="Green">E18</font></i>: 
( <span class="p1">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="cat_2.html#E2:28_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="cat_2.html#T22" target="_self">Th22</a></i>;<br/><b>assume </b><a NAME="E4:28_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="cat_2.html#E3:28_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E7:28_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:28_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="cat_2.html#E3:28_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="cat_2.html#E4:28_1_1"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E8:28_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="cat_2.html#E1:28_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="cat_2.html#E3:28_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="cat_2.html#E4:28_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="cat_2.html#E7:28_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E9:28_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:28_1"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:28"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/> <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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_1.html#D23">CAT_1:def 23</a><br/>

<b>assume </b><a NAME="E3:28"/><i><font color="Green">E17</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>5</sub></font>;<br/>

<a NAME="E4:28"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="cat_2.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E5:28"/><b>then </b><i><font color="Green">E19</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_2.html#E2:28"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:28"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="cat_2.html#E2:28"><i><font color="Green">E16</font></i></a>, <a class="txt" href="cat_2.html#E3:28"><i><font color="Green">E17</font></i></a>, <a class="txt" href="cat_2.html#E4:28"><i><font color="Green">E18</font></i></a></i>;<br/>

<a NAME="E7:28"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_2.html#E3:28"><i><font color="Green">E17</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="cat_2.html#E5:28"><i><font color="Green">E19</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E9:28"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K8" target="_self">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>


</div>
