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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D1</font>, <font color="Olive" title="b2">D2</font> being <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><br/> for <font color="Olive" title="b3">F1</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">D1</font><br/> for <font color="Olive" title="b4">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">D2</font>  st <font color="Olive" title="b3">F1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b4">F2</font> holds <br/> <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Olive" title="b3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Olive" title="b4">F2</font></span><br/><b>let </b><font color="Maroon" title="c2">D1</font>, <font color="Maroon" title="c3">D2</font> be  <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F1</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D1</font><br/> for <font color="Olive" title="b2">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">D2</font>  st <font color="Olive" title="b1">F1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">F2</font> holds <br/> <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Olive" title="b2">F2</font></span><br/><b>let </b><font color="Maroon" title="c4">F1</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">D2</font>  st <font color="Maroon" title="c4">F1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">F2</font> holds <br/> <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Olive" title="b1">F2</font></span><br/><b>let </b><font color="Maroon" title="c5">F2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">D2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">F1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">F2</font> implies  <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Maroon" title="c5">F2</font> )</span><br/>







<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Maroon" title="c5">F2</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">DD</font> = the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">D1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">D2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E3:45"/>
<font color="Maroon" title="c6">DD</font> is <a href="cat_5.html#V2" title="CAT_5:attr.2">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" title="c7">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">DD</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D5" target="_self" title="CAT_5:def.5">CAT_5:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">d</font> is   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a></span><br/>

<a NAME="E1:45_1"/>
( <font color="Maroon" title="c7">d</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D1</font> or <font color="Maroon" title="c7">d</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">D2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:45_1"/>
<font color="Maroon" title="c7">d</font> is   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>
 <b>by </b><i><a class="ref" href="cat_5.html#T12" target="_self" title="CAT_5:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">DD</font> = the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">D1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">D2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="cat_5.html#V2" title="CAT_5:attr.2">categorial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c8">D</font> being  <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a> <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a> <a href="cat_5.html#V4" title="CAT_5:attr.4">full</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><b> such that </b><br/><a NAME="E6:45"/><i><font color="Green" title="E29">A2</font></i>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c8">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">DD</font>
 <b>by </b><i><a class="ref" href="cat_5.html#T20" target="_self" title="CAT_5:th.20">Th20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">D1</font> = <font color="Maroon" title="c2">D1</font>, <font color="Maroon" title="c10">D2</font> = <font color="Maroon" title="c3">D2</font> as    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c8">D</font> <b>by </b><i><a class="txt" href="cat_5.html#E6:45"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="cat_5.html#T21" target="_self" title="CAT_5:th.21">Th21</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">F1</font> = <font color="Maroon" title="c4">F1</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c9">D1</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">F2</font> = <font color="Maroon" title="c5">F2</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">D2</font> ;<br/>
<a NAME="E10:45"/>
(  <a href="cat_2.html#K8" title="CAT_2:func.8">incl</a> <font color="Maroon" title="c9">D1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c9">D1</font> &amp;  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c9">D1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c9">D1</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c9">D1</font> )
 
;<br/>
<a NAME="E11:45"/><b>then </b>
<font color="Maroon" title="c11">F1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K8" title="CAT_2:func.8">incl</a> <font color="Maroon" title="c9">D1</font></span>)</span> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c11">F1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">G1</font> = <font color="Maroon" title="c11">F1</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c8">D</font> ;<br/>
 <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Maroon" title="c11">F1</font> = 
 <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Maroon" title="c13">G1</font>
<b>by </b><i><a class="ref" href="cat_5.html#T9" target="_self" title="CAT_5:th.9">Th9</a></i>
<br/>.= 
 <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Maroon" title="c12">F2</font>
<b>by </b><i><a class="txt" href="cat_5.html#E1:45"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="cat_5.html#T9" target="_self" title="CAT_5:th.9">Th9</a></i>
;<br/>
<b>hence </b><a NAME="E14:45"/>
 <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <font color="Maroon" title="c5">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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