<?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>thus </b><a NAME="E1:30"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="cat_2.html#R1" target="_self">is_full_subcategory_of</a> <font color="Maroon">c<sub>1</sub></font> implies  <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="cat_2.html#R1" target="_self">is_full_subcategory_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:30_1"/><b>then </b>
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>
 
<b>by </b><i><a class="ref" href="cat_2.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E3:30_1"/>
 <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>
 <b>by </b><i><a class="ref" href="cat_2.html#T25" target="_self">Th25</a></i>;<br/>


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

<b>assume </b><a NAME="E2:30"/>
 <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>
 ;<br/>

<b>hence </b><a NAME="E3:30"/>
( <font color="Maroon">c<sub>2</sub></font> is    <a href="cat_2.html#M3" target="_self">Subcategory</a> of <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_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> ) )
 <b>by </b><i><a class="ref" href="cat_2.html#T25" target="_self">Th25</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_2.html#D6" target="_self">CAT_2:def 6</a><br/>


</div>
