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<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">D</font> be    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">E</font> be    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">D</font>;<br/>





<a NAME="E1:12"/><i><font color="Green">E17</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font><br/> for <font color="Olive">a'</font>, <font color="Olive">b'</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a'</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b'</font> holds <br/> <a href="cat_1.html#K6">Hom</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">a'</font>,<font color="Olive">b'</font> ) &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> &amp; ( for <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font><br/> for <font color="Olive">a'</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a'</font> holds <br/> <a href="cat_1.html#K10">id</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">a'</font> ) )
 
<b>by </b><i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a></i>;<br/>
<a NAME="E2:12"/><i><font color="Green">E40</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">D</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">E</font><br/> for <font color="Olive">a'</font>, <font color="Olive">b'</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a'</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b'</font> holds <br/> <a href="cat_1.html#K6">Hom</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">a'</font>,<font color="Olive">b'</font> ) &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">E</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">a'</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">E</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a'</font> holds <br/> <a href="cat_1.html#K10">id</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">a'</font> ) )
 
<b>by </b><i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:12"/>
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">E</font>, <font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">E</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">b'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="cat_2.html#T12">CAT_2:12</a></i>;<br/><b>assume </b><a NAME="E2:12_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> )
 ;<br/><a NAME="E3:12_1"/><b>then </b>
(  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> )
 <b>by </b><i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a></i>;<br/><b>hence </b><a NAME="E4:12_1"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E5:12"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">E</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">E</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/>



<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="cat_2.html#T12">CAT_2:12</a></i>;<br/>
<b>assume </b><a NAME="E7:12"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/>

<a NAME="E8:12"/><b>then </b>
(  <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">a1</font> &amp;  <a href="cat_1.html#K10">id</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">a'</font> )
 
<b>by </b><i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:12"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">a'</font>
 ;<br/>


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