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<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>, <font color="Maroon">D</font> be   <a href="cat_1.html#NM4">Category</a>;<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>let </b><font color="Maroon">F</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">C</font>,<font color="Maroon">E</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/>







<b>assume </b><a NAME="E1:19"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">G</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">S</font> =  <a href="cat_5.html#K7" target="_self">Image</a> <font color="Maroon">F</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/>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:19_1"/>
(  <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> &amp;  <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">G</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/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:19_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> ;<br/><b>reconsider </b><font color="Maroon">b</font> = <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> 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/><font color="Maroon">G</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">a</font></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T104">CAT_1:104</a></i>
<br/>.= 
 <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="E6:19_1"/>
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <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="cat_1.html#T103">CAT_1:103</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:19"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font> <a href="funct_2.html#R1">=</a>  <a href="cat_1.html#K12">Obj</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E5:19"/><b>then </b><i><font color="Green">E46</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">S</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">G</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">S</font> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">T</font> be    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:19_2"/><i><font color="Green">E47</font></i>: 
( 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">G</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">T</font> )
 ;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><a NAME="E3:19_2"/><i><font color="Green">E48</font></i>: 
( <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <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_2.html#K3">.</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/><a NAME="E4:19_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">T</font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:19_2"/><b>then </b><i><font color="Green">E49</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">T</font> <a href="subset_1.html#R2">meets</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>then reconsider </b><font color="Maroon">E1</font> = <font color="Maroon">T</font> <a href="cat_5.html#K6" target="_self">/\</a> <font color="Maroon">E</font> as    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/><a NAME="E7:19_2"/>
(  <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> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">E</font> &amp; the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">E1</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">T</font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">E</font> )
 <b>by </b><i>, <a class="txt" href="cat_5.html#E1:19"><i><font color="Green">E17</font></i></a>, </i>;<br/><a NAME="E8:19_2"/><b>then </b><i><font color="Green">E51</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">E1</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>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E9:19_2"/>
( the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">E1</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">T</font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">E</font> &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> )
 <b>by </b><i>, <a class="txt" href="cat_5.html#E1:19"><i><font color="Green">E17</font></i></a>, </i>;<br/><a NAME="E10:19_2"/><b>then </b>
 <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">E1</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E11:19_2"/><b>then </b>
(  <a href="cat_5.html#K7" target="_self">Image</a> <font color="Maroon">F</font> is    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">E1</font> &amp; <font color="Maroon">E1</font> is    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">T</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E12:19_2"/>
<font color="Maroon">S</font> is    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">T</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:19"/>
 <a href="cat_5.html#K7" target="_self">Image</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="cat_5.html#K7" target="_self">Image</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="cat_5.html#T1" target="_self">Th1</a>, </i>;<br/>


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