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<b>let </b><font color="Maroon" title="c1">C1</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c2">C2</font> be    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c1">C1</font>;<br/>



<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c1">C1</font> is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c2">C2</font>
 ;<br/>

<a NAME="E2:11"/><i><font color="Green" title="E6">A2</font></i>: 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">C2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:11_1"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">C2</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:11"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:11_1"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C1</font>
 <b>by </b><i><a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:11"/><i><font color="Green" title="E7">A3</font></i>: 
the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">C2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:11_2"/>
the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">C2</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:11"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="cat_2.html#T13" title="CAT_2:th.13">CAT_2:13</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:11_2"/>
the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C1</font>
 <b>by </b><i><a class="ref" href="cat_2.html#T13" title="CAT_2:th.13">CAT_2:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:11"/><i><font color="Green" title="E8">A4</font></i>: 
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">C2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:11_3"/>
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">C2</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:11"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:11_3"/>
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C1</font>
 <b>by </b><i><a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">m1</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C1</font>;<br/><b>reconsider </b><font color="Maroon" title="c4">m2</font> = <font color="Maroon" title="c3">m1</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C2</font> <b>by </b><i><a class="txt" href="cat_5.html#E3:11"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/><b>thus </b>the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">m1</font> = 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c3">m1</font>

<br/>.= 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c4">m2</font>
<b>by </b><i><a class="ref" href="cat_2.html#T15" title="CAT_2:th.15">CAT_2:15</a></i>
<br/>.= 
the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m1</font>

;<br/></div><b>end;</b></div>
<a NAME="E6:11"/><b>then </b><i><font color="Green" title="E9">A5</font></i>: 
the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">C2</font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:11"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="cat_5.html#E3:11"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">m1</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C1</font>;<br/><b>reconsider </b><font color="Maroon" title="c4">m2</font> = <font color="Maroon" title="c3">m1</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C2</font> <b>by </b><i><a class="txt" href="cat_5.html#E3:11"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/><b>thus </b>the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">m1</font> = 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c3">m1</font>

<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c4">m2</font>
<b>by </b><i><a class="ref" href="cat_2.html#T15" title="CAT_2:th.15">CAT_2:15</a></i>
<br/>.= 
the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m1</font>

;<br/></div><b>end;</b></div>
<a NAME="E8:11"/><b>then </b><i><font color="Green" title="E10">A6</font></i>: 
the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">C2</font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:11"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="cat_5.html#E3:11"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">o1</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C1</font>;<br/><b>reconsider </b><font color="Maroon" title="c4">o2</font> = <font color="Maroon" title="c3">o1</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">C2</font> <b>by </b><i><a class="txt" href="cat_5.html#E2:11"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/><b>thus </b>the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">o1</font> = 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">o1</font>

<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">o2</font>
<b>by </b><i><a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a></i>
<br/>.= 
the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">o1</font>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:11"/>
 <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C1</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C1</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C1</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C1</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C1</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">C2</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">C2</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">C2</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">C2</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">C2</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">C2</font> #)
 <b>by </b><i><a class="txt" href="cat_5.html#E2:11"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="cat_5.html#E3:11"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="cat_5.html#E4:11"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="cat_5.html#E6:11"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="cat_5.html#E8:11"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
