<?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="E1:14"/>
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> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = 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> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <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>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <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>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E4:14"/>
( <font color="Maroon">c<sub>5</sub></font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:14"/><b>then </b><i><font color="Green">E8</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="cat_1.html#U6">Id</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="cat_1.html#U6">Id</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>3</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> &amp; the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>2</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="cat_5.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E6:14"/><i><font color="Green">E9</font></i>: 
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="nattra_1.html#T8">NATTRA_1:8</a></i>;<br/>
<b>then </b><i><font color="Green">E10</font></i>: the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><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></span>)</span> = 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E5:14"><i><font color="Green">E8</font></i></a>, <a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><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></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E5:14"><i><font color="Green">E8</font></i></a>, <a class="txt" href="cat_5.html#E6:14"><i><font color="Green">E9</font></i></a>, <a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
;<br/>
<i><font color="Green">E11</font></i>: the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><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></span>)</span> = 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E5:14"><i><font color="Green">E8</font></i></a>, <a class="txt" href="cat_5.html#E6:14"><i><font color="Green">E9</font></i></a>, <a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><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></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E5:14"><i><font color="Green">E8</font></i></a>, <a class="txt" href="cat_5.html#E6:14"><i><font color="Green">E9</font></i></a>, <a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
;<br/>
<i><font color="Green">E12</font></i>: the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><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></span>)</span> = 
<span class="p1">(<span class="default">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></span>)</span> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E5:14"><i><font color="Green">E8</font></i></a>, <a class="ref" href="nattra_1.html#T8">NATTRA_1:8</a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</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>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</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>3</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default">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>3</sub></font></span>)</span> <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="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><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></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E5:14"><i><font color="Green">E8</font></i></a>, <a class="ref" href="nattra_1.html#T8">NATTRA_1:8</a></i>
;<br/>
the <a href="cat_1.html#U6">Id</a> of <span class="p1">(<span class="default"><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></span>)</span> = 
<span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E5:14"><i><font color="Green">E8</font></i></a>, <a class="ref" href="nattra_1.html#T8">NATTRA_1:8</a></i>
<br/>.= 
the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">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></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</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="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
the <a href="cat_1.html#U6">Id</a> of <span class="p1">(<span class="default"><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></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E5:14"><i><font color="Green">E8</font></i></a>, <a class="ref" href="nattra_1.html#T8">NATTRA_1:8</a></i>
;<br/>
<b>hence </b><a NAME="E11:14"/>
<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#E5:14"><i><font color="Green">E8</font></i></a>, <a class="txt" href="cat_5.html#E7:14"><i><font color="Green">E10</font></i></a>, <a class="txt" href="cat_5.html#E8:14"><i><font color="Green">E11</font></i></a>, <a class="txt" href="cat_5.html#E9:14"><i><font color="Green">E12</font></i></a></i>;<br/>


</div>
