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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="cat_1.html#V1">strict</a> <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="cat_1.html#V1">strict</a>  <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green">E11</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:13"/><i><font color="Green">E12</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>thus </b>the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>2</sub></font> = 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K7">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="nattra_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E1:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="nattra_1.html#E2:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>
;<br/><b>thus </b>the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>2</sub></font> = 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K7">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="nattra_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E1:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="nattra_1.html#E2:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>
;<br/><a NAME="E3:13_1"/><i><font color="Green">E13</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>thus </b>the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>2</sub></font> = 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="nattra_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E1:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="nattra_1.html#E3:13_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>
;<br/><b>thus </b>the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>2</sub></font> = 
the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K7">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="nattra_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E1:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="nattra_1.html#E2:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:13"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E1:13"><i><font color="Green">E11</font></i></a></i>;<br/>


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