<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:15"/><i><font color="Green">E9</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#R2">meets</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:15"/><b>then </b><i><font color="Green">E10</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E3:15"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:15"><i><font color="Green">E9</font></i></a>, <a class="ref" href="cat_5.html#T5" target="_self">Th5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E12</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:15_1"/><i><font color="Green">E13</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>4</sub></font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>5</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="cat_5.html#E1:15_1"><i><font color="Green">E12</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E5:15_1"/>
( <font color="Maroon">c<sub>7</sub></font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:15_1"/><b>then </b><i><font color="Green">E14</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>5</sub></font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>5</sub></font> &amp; the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>5</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>5</sub></font> &amp; the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>5</sub></font> &amp; the <a href="cat_1.html#U6">Id</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="cat_5.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E7:15_1"/>
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/><a NAME="E8:15_1"/><b>then </b><i><font color="Green">E15</font></i>: 
the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_5.html#E6:15_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>;<br/><a NAME="E9:15_1"/>
( the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/><a NAME="E10:15_1"/><b>then </b><i><font color="Green">E16</font></i>: 
the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_5.html#E6:15_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>;<br/><a NAME="E11:15_1"/>
( the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/><a NAME="E12:15_1"/><b>then </b><i><font color="Green">E17</font></i>: 
the <a href="cat_1.html#U6">Id</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_5.html#E6:15_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>;<br/><a NAME="E13:15_1"/>
 <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</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>4</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E14:15_1"/><b>then </b>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/><a NAME="E15:15_1"/><b>then </b>
the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2"><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>4</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p2"><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>5</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>5</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="cat_5.html#E6:15_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>;<br/><a NAME="E16:15_1"/><b>then </b>
the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_5.html#E6:15_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>;<br/><b>hence </b><a NAME="E17:15_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cat_5.html#E2:15_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_5.html#E6:15_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="cat_5.html#E8:15_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cat_5.html#E10:15_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="cat_5.html#E12:15_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nattra_1.html#T9">NATTRA_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:15"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:15"><i><font color="Green">E10</font></i></a>, <a class="txt" href="cat_5.html#E3:15"><i><font color="Green">E11</font></i></a></i>;<br/>


</div>
