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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">E</font> being   <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b1">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> iff  <a href="cat_2.html#K8" title="CAT_2:func.8">incl</a> <font color="Olive" title="b1">E</font> is  <a href="cat_1.html#V9" title="CAT_1:attr.9">full</a>  )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> iff  <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>  )</span><br/>



<span class="kw">thus </span><a NAME="E1:39"/>
( <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>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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>  implies <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> </span><br/>

<a NAME="E2:39_1"/><span class="kw">then </span>
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">a9</font>, <font color="Olive" title="b4">b9</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">a9</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b9</font> holds <br/> <a href="cat_1.html#K3" title="CAT_1:func.3">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#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b3">a9</font>,<font color="Olive" title="b4">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#D6" target="_self" title="CAT_2:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T25" target="_self" title="CAT_2:th.25">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:39"/>
 <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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">hence </span><a NAME="E3:39"/>
( <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">a9</font>, <font color="Olive" title="b4">b9</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">a9</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b9</font> holds <br/> <a href="cat_1.html#K3" title="CAT_1:func.3">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#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b3">a9</font>,<font color="Olive" title="b4">b9</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T25" target="_self" title="CAT_2:th.25">Th25</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="cat_2.html#D6" target="_self" title="CAT_2:def.6">CAT_2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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