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

<b>consider </b><font color="Maroon">S</font> being    <a href="index_1.html#M4" target="_self">TargetCat</a> of <font color="Maroon">I</font>;<br/>
<b>reconsider </b><font color="Maroon">T</font> =  <a href="cat_5.html#K7">Image</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K15" target="_self">-functor</a> <font color="Maroon">C</font>,<font color="Maroon">S</font></span>)</span> as  <a href="cat_1.html#V1">strict</a>  <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">S</font> ;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">I</font> <a href="index_1.html#K15" target="_self">-functor</a> <font color="Maroon">C</font>,<font color="Maroon">S</font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">C</font>,<font color="Maroon">T</font> <b>by </b><i><a class="ref" href="cat_5.html#T8">CAT_5:8</a></i>;<br/>
<a NAME="E4:47_1_2"/>
<font color="Maroon">T</font> is    <a href="index_1.html#M4" target="_self">TargetCat</a> of <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:47_1_2_1"/>
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K15" target="_self">-functor</a> <font color="Maroon">C</font>,<font color="Maroon">S</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_5.html#D3">CAT_5:def 3</a></i>;<br/>
<a NAME="E2:47_1_2_1"/><b>then </b>
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K10" target="_self">`1</a> </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="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1_2_1"/>
for <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> holds  <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <font color="Olive">a</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="index_1.html#D9" target="_self">INDEX_1:def 9</a><br/>

<b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E4:47_1_2_1"/>
( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">b</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:47_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <span class="p4">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <span class="p4">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">T</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</font> as  <a href="cat_1.html#V1">strict</a>  <a href="index_1.html#M4" target="_self">TargetCat</a> of <font color="Maroon">I</font> ;<br/>
<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>let </b><font color="Maroon">T'</font> be    <a href="index_1.html#M4" target="_self">TargetCat</a> of <font color="Maroon">I</font>;<br/>

<a NAME="E6:47_1_2"/>
<font color="Maroon">I</font> <a href="index_1.html#K15" target="_self">-functor</a> <font color="Maroon">C</font>,<font color="Maroon">S</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">I</font> <a href="index_1.html#K15" target="_self">-functor</a> <font color="Maroon">C</font>,<font color="Maroon">T'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:47_1_2"/>
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="cat_5.html#K7">Image</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K15" target="_self">-functor</a> <font color="Maroon">C</font>,<font color="Maroon">T'</font></span>)</span>
 <b>by </b><i><a class="ref" href="cat_5.html#T22">CAT_5:22</a></i>;<br/>


</div>
