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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c2">E</font> be    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c1">C</font>;<br/>



<b>thus </b><a NAME="E1:30"/>
( <font color="Maroon" title="c2">E</font> <a href="cat_2.html#R1" title="CAT_2:pred.1">is_full_subcategory_of</a> <font color="Maroon" title="c1">C</font> implies  <a href="cat_2.html#K8" title="CAT_2:func.8">incl</a> <font color="Maroon" title="c2">E</font> is <a href="cat_1.html#V9" title="CAT_1:attr.9">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" title="c2">E</font> <a href="cat_2.html#R1" title="CAT_2:pred.1">is_full_subcategory_of</a> <font color="Maroon" title="c1">C</font>
 ;<br/>

<a NAME="E2:30_1"/><b>then </b>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">E</font><br/> for <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b'</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a'</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b'</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b4">b'</font>
 
<b>by </b><i><a class="ref" href="cat_2.html#D6" target="_self" title="CAT_2:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E3:30_1"/>
 <a href="cat_2.html#K8" title="CAT_2:func.8">incl</a> <font color="Maroon" title="c2">E</font> is <a href="cat_1.html#V9" title="CAT_1:attr.9">full</a>
 <b>by </b><i><a class="ref" href="cat_2.html#T25" target="_self" title="CAT_2:th.25">Th25</a></i>;<br/>


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

<b>assume </b><a NAME="E2:30"/>
 <a href="cat_2.html#K8" title="CAT_2:func.8">incl</a> <font color="Maroon" title="c2">E</font> is <a href="cat_1.html#V9" title="CAT_1:attr.9">full</a>
 ;<br/>

<b>hence </b><a NAME="E3:30"/>
( <font color="Maroon" title="c2">E</font> is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c1">C</font> &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">E</font><br/> for <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b'</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a'</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b'</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b4">b'</font> ) )
 <b>by </b><i><a class="ref" href="cat_2.html#T25" target="_self" title="CAT_2:th.25">Th25</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_2.html#D6" target="_self" title="CAT_2:def.6">CAT_2:def 6</a><br/>


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