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

<b>consider </b><font color="Maroon">a</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/>
<a NAME="E2:16_1_1"/><i><font color="Green">E17</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:16_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">O</font> =  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">O</font> = <font color="Maroon">O</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">D</font> ;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>
<a NAME="E7:16_1_1"/><i><font color="Green">E40</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> &amp;  ex <font color="Olive">E</font> being   <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">D</font> st <br/>( the <a href="cat_1.html#U1">Objects</a> of <font color="Olive">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <font color="Olive">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<b>consider </b><font color="Maroon">MM</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:16_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MM</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>set </b><font color="Maroon">HH</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Olive">a0</font>,<font color="Olive">b</font></span>)</span> where <font color="Olive">a0</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>, <font color="Olive">b</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font> : ( <font color="Olive">a0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> ) </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">M0</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Olive">a0</font>,<font color="Olive">b</font></span>)</span> where <font color="Olive">a0</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>, <font color="Olive">b</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font> : ( <font color="Olive">a0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="cat_2.html#T28">CAT_2:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D0</font> = the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M0</font>, <font color="Maroon">C0</font> = the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M0</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M0</font>,<font color="Maroon">O</font> <b>by </b><i><a class="ref" href="cat_2.html#T29">CAT_2:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">CC</font> = the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="realset1.html#K1">||</a> <font color="Maroon">M0</font> as   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">M0</font>,<font color="Maroon">M0</font></span><a href="mcart_1.html#K12">:]</a></span>,<font color="Maroon">M0</font> <b>by </b><i><a class="ref" href="cat_2.html#T29">CAT_2:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">I0</font> = the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">O</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">O</font>,<font color="Maroon">M0</font> <b>by </b><i><a class="ref" href="cat_2.html#T29">CAT_2:29</a></i>;<br/>
<a NAME="E14:16_1_1"/>
 <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M0</font>,<font color="Maroon">D0</font>,<font color="Maroon">C0</font>,<font color="Maroon">CC</font>,<font color="Maroon">I0</font> #) <a href="cat_2.html#R1">is_full_subcategory_of</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="cat_2.html#T30">CAT_2:30</a></i>;<br/>
<a NAME="E15:16_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M0</font>,<font color="Maroon">D0</font>,<font color="Maroon">C0</font>,<font color="Maroon">CC</font>,<font color="Maroon">I0</font> #) is    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="cat_2.html#D6">CAT_2:def 6</a></i>;<br/>
<a NAME="E16:16_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E5:16_1_1_1"/>
( <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:16_1_1_1"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T105">CAT_1:105</a></i>;<br/>
<a NAME="E7:16_1_1_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Olive">a0</font>,<font color="Olive">b</font></span>)</span> where <font color="Olive">a0</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>, <font color="Olive">b</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font> : ( <font color="Olive">a0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> ) </span>}</span>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:16_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M0</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:16_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">M0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MM</font>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">MM</font>;<br/>
<a NAME="E18:16_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MM</font> holds <br/> <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font>
 
<b>by </b><i/>;<br/>
<a NAME="E19:16_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">MM</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:16_1_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MM</font>
 ;<br/><a NAME="E2:16_1_1_2"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:16_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">MM</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E22:16_1_1"/>
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:16_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M0</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font> ;<br/>
<b>set </b><font color="Maroon">DOM</font> = the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M</font>;<br/>
<b>set </b><font color="Maroon">COD</font> = the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M</font>;<br/>
<b>set </b><font color="Maroon">COMP</font> = the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="realset1.html#K1">||</a> <font color="Maroon">M</font>;<br/>
<b>set </b><font color="Maroon">ID</font> = the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">O</font>;<br/>
<a NAME="E24:16_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E25:16_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1_1_4"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font> <b>by </b><i><a class="txt" href="cat_5.html#E2:16_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E5:16_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M0</font>
 
<b>by </b><i>, <a class="txt" href="cat_5.html#E2:16_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:16_1_1_4"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Olive">a0</font>,<font color="Olive">b</font></span>)</span> where <font color="Olive">a0</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>, <font color="Olive">b</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font> : ( <font color="Olive">a0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>, <font color="Maroon">b</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E9:16_1_1_4"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:16_1_1_4"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="txt" href="cat_5.html#E2:16_1_1"><i><font color="Green">E17</font></i></a>, , , <a class="ref" href="cat_1.html#T18">CAT_1:18</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E11:16_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">DOM</font> = the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">O</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E27:16_1_1"/><i><font color="Green">E64</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E28:16_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1_1_5"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:16_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font> <b>by </b><i/>;<br/>
<a NAME="E5:16_1_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M0</font>
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:16_1_1_5"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Olive">a0</font>,<font color="Olive">b</font></span>)</span> where <font color="Olive">a0</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>, <font color="Olive">b</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font> : ( <font color="Olive">a0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>, <font color="Maroon">b</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E9:16_1_1_5"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:16_1_1_5"/>
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:16_1_1"><i><font color="Green">E17</font></i></a>, , , , <a class="ref" href="cat_1.html#T18">CAT_1:18</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E11:16_1_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">COD</font> = the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">M</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">O</font> <b>by </b><i><a class="txt" href="cat_5.html#E2:16_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E30:16_1_1"/><i><font color="Green">E70</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="realset1.html#K1">||</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T87">RELAT_1:87</a></i>;<br/>
<a NAME="E31:16_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="realset1.html#K1">||</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="realset1.html#K1">||</a> <font color="Maroon">M</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1_1_6"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="realset1.html#K1">||</a> <font color="Maroon">M</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="realset1.html#K1">||</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">x2</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">M</font> <b>by </b><i>, , <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E5:16_1_1_6"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="cat_2.html#K13">]</a></span>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:16_1_1_6_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MM</font>
 ;<br/><b>then consider </b><font color="Maroon">E</font> being    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E3:16_1_1_6_1"/><i><font color="Green">E76</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">x1</font>, <font color="Maroon">y2</font> = <font color="Maroon">x2</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">E</font> <b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E5:16_1_1_6_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="realset1.html#K1">||</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E6:16_1_1_6_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:16_1_1_6_1"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">x2</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">x1</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">x2</font> )
 <b>by </b><i>, <a class="ref" href="cat_1.html#T40">CAT_1:40</a>, <a class="ref" href="cat_2.html#T15">CAT_2:15</a></i>;<br/><a NAME="E8:16_1_1_6_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>;<br/><a NAME="E9:16_1_1_6_1"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a></i>;<br/><b>then </b>the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/><a NAME="E11:16_1_1_6_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">E</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E12:16_1_1_6_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:16_1_1_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">COMP</font> = the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="realset1.html#K1">||</a> <font color="Maroon">M</font> as   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="mcart_1.html#K12">:]</a></span>,<font color="Maroon">M</font> <b>by </b><i>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E33:16_1_1"/><i><font color="Green">E78</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">O</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E34:16_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">O</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">O</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1_1_7"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">O</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:16_1_1_7"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x'</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<font color="Maroon">y</font> = 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T104">CAT_1:104</a></i>
;<br/>
<a NAME="E9:16_1_1_7"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:16_1_1_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="cat_5.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ID</font> = the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">O</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">O</font>,<font color="Maroon">M</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T</font> =  <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) as  <a href="cat_1.html#V1">strict</a>  <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="nattra_1.html#T9">NATTRA_1:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>thus </b><a NAME="E37:16_1_1"/>
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">T</font> )
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<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/>

<b>assume </b><a NAME="E38:16_1_1"/><i><font color="Green">E83</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">E</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">E</font> )
 ;<br/>

<b>hence </b><a NAME="E39:16_1_1"/>
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">E</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a><br/>

<a NAME="E40:16_1_1"/>
the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="cat_2.html#T13">CAT_2:13</a></i>;<br/>
<a NAME="E41:16_1_1"/><b>then </b>
the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MM</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E42:16_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<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">T</font>, <font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">T</font>;<br/><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>assume </b><a NAME="E1:16_1_1_8"/><i><font color="Green">E88</font></i>: 
( <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/><b>thus </b><a NAME="E2:16_1_1_8"/>
 <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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:16_1_1_8_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">T</font> : (  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:16_1_1_8_1"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">E</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:16_1_1_8_1"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, <a class="ref" href="cat_2.html#T15">CAT_2:15</a></i>;<br/>
<a NAME="E8:16_1_1_8_1"/><b>then </b>
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="cat_2.html#T15">CAT_2:15</a></i>;<br/>
<b>hence </b><a NAME="E9:16_1_1_8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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_5.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<a NAME="E44:16_1_1"/><i><font color="Green">E92</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_9"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">T</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">x2</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">x1</font>, <font color="Maroon">y2</font> = <font color="Maroon">x2</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">x1</font>, <font color="Maroon">z2</font> = <font color="Maroon">x2</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">E</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:16_1_1_9"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="cat_2.html#K13">]</a></span>
 
<b>by </b><i><a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E6:16_1_1_9"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">y2</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">x1</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">x1</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">x2</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">x2</font> )
 
<b>by </b><i><a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, <a class="ref" href="cat_1.html#T40">CAT_1:40</a>, <a class="ref" href="cat_2.html#T15">CAT_2:15</a></i>;<br/>
<b>hence </b><a NAME="E7:16_1_1_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a>, <a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:16_1_1_10"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">T</font>
 ;<br/><a NAME="E2:16_1_1_10"/><i><font color="Green">E96</font></i>: 
( the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">T</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; 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> )
 <b>by </b><i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a></i>;<br/><b>hence </b>the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, <a class="ref" href="cat_5.html#T2" target="_self">Th2</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E46:16_1_1"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">T</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">T</font>;<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>reconsider </b><font color="Maroon">b</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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>assume </b><a NAME="E48:16_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/>

<a NAME="E49:16_1_1"/><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">b</font> &amp;  <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">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="E50:16_1_1"/>
 <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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