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

<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<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>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">a</font>,<font color="Maroon">I</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:66"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="cat_3.html#R1" target="_self">is_a_product_wrt</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E2:66"/><i><font color="Green">E42</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>and </b><br/><a NAME="E3:66"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> is <a href="cat_1.html#V5">invertible</a>
 ;<br/>

<b>thus </b><a NAME="E4:66"/>
<font color="Maroon">F</font> <a href="cat_3.html#K7" target="_self">*</a> <font color="Maroon">f</font> is    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">b</font>,<font color="Maroon">I</font>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D15" target="_self">CAT_3:def 15</a><br/>

<b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">F'</font> be    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">c</font>,<font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E5:66"/><i><font color="Green">E60</font></i>: 
 <a href="cat_3.html#K4" target="_self">cods</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_3.html#K7" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">F'</font>
 ;<br/>

<a NAME="E6:66"/>
 <a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">I</font> <a href="cat_3.html#K1" target="_self">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:66"/><b>then </b>
 <a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">F'</font>
 
<b>by </b><i><a class="txt" href="cat_3.html#E3:66"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">h</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E9:66"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E10:66"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">k</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">a</font> holds <br/>( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font> ) iff <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E12:66"/><i><font color="Green">E78</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> )
 <b>and </b><br/><a NAME="E13:66"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E14:66"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_3.html#E2:66"><i><font color="Green">E42</font></i></a>, <a class="ref" href="cat_1.html#D12">CAT_1:def 12</a></i>;<br/>
<b>take </b><font color="Maroon">gh</font> = <font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font>;<br/>

<a NAME="E15:66"/><i><font color="Green">E98</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self">Th1</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E16:66"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/>
<b>hence </b><a NAME="E17:66"/>
<font color="Maroon">gh</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>
 ;<br/>

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

<b>assume </b><a NAME="E18:66"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E19:66"/><b>then </b><i><font color="Green">E99</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<b>thus </b><a NAME="E20:66"/>
( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_3.html#K7" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font> ) implies <font color="Maroon">gh</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_3.html#K7" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>now </b></a><div class="add"><a NAME="E1:66_1_1"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, , <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/><b>hence </b><a NAME="E2:66_1_1"/>
<font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>
 ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:66_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E4:66_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="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_3.html#K7" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="cat_3.html#D3" target="_self">Def3</a>, </i>
<br/>.= 
<font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D3" target="_self">Def3</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E3:66_1"/><b>then </b>
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">gh</font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="cat_3.html#T2" target="_self">Th2</a>, , <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>
;<br/><br/>


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

<b>assume </b><a NAME="E21:66"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">gh</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E22:66"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E23:66"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_3.html#K7" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="cat_3.html#D3" target="_self">Def3</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="cat_3.html#D4" target="_self">Def4</a>, <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="cat_3.html#T7" target="_self">Th7</a>, <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#T7" target="_self">Th7</a>, <a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>
<br/>.= 
<font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self">Th1</a>, , <a class="ref" href="cat_3.html#D3" target="_self">Def3</a></i>
;<br/>


</div>
