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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c2">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>;<br/>







<b>set </b><font color="Maroon" title="c5">a</font> =  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font>;<br/>
<b>set </b><font color="Maroon" title="c6">b</font> =  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">p2</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:75"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c2">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 <b>and </b><br/><a NAME="E2:75"/><i><font color="Green" title="E44">A2</font></i>: 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">p2</font> is <a href="cat_1.html#V7" title="CAT_1:attr.7">terminal</a>
 ;<br/>

<a NAME="E3:75"/><i><font color="Green" title="E45">A3</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:75"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="cat_3.html#D16" target="_self" title="CAT_3:def.16">Def16</a></i>;<br/>
<b>hence </b><a NAME="E4:75"/>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">c</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_1.html#T19" title="CAT_1:th.19">CAT_1:19</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D17" title="CAT_1:def.17">CAT_1:def 17</a><br/>

<b>reconsider </b><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c3">p1</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">c</font>, <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font> <b>by </b><i><a class="txt" href="cat_3.html#E3:75"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="cat_1.html#T22" title="CAT_1:th.22">CAT_1:22</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">p</font>
;<br/>

<b>set </b><font color="Maroon" title="c8">f</font> =  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">g</font> =  <a href="cat_3.html#K9" title="CAT_3:func.9">term</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span>;<br/>
<a NAME="E6:75"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_3.html#K9" title="CAT_3:func.9">term</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="cat_3.html#K9" title="CAT_3:func.9">term</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">p2</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E2:75"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="cat_3.html#T39" target="_self" title="CAT_3:th.39">Th39</a></i>;<br/>
<a NAME="E7:75"/><b>then </b>
(  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span> &amp;  <a href="cat_3.html#K9" title="CAT_3:func.9">term</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E9:75"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span>,<font color="Maroon" title="c2">c</font>
 <b>and </b><br/><a NAME="E10:75"/><i><font color="Green" title="E47">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span>,<font color="Maroon" title="c2">c</font> holds <br/>( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_3.html#K9" title="CAT_3:func.9">term</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span> ) iff <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:75"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="cat_3.html#D16" target="_self" title="CAT_3:def.16">Def16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">h</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D12" title="CAT_1:def.12">CAT_1:def 12</a><br/>

<b>thus </b><a NAME="E11:75"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">p</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">p</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E3:75"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="cat_3.html#E9:75"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>

<b>thus </b><a NAME="E12:75"/><i><font color="Green" title="E48">A6</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_3.html#E9:75"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="cat_3.html#E10:75"><i><font color="Green" title="E47">A5</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>now </b></a><div class="add"><a NAME="E1:75_1"/><i><font color="Green" title="E49">A7</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">p1</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E9:75"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/><a NAME="E2:75_1"/><b>then </b><i><font color="Green" title="E50">A8</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E3:75"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/><b>hence </b><a NAME="E3:75_1"/>
<font color="Maroon" title="c10">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c2">c</font>
 ;<br/><b>thus </b><font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">p</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">p</font>
<b>by </b><i><a class="txt" href="cat_3.html#E3:75"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="cat_3.html#E12:75"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="cat_3.html#E1:75_1"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>
<br/>.= 
<font color="Maroon" title="c7">p</font>
<b>by </b><i><a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a></i>
;<br/><a NAME="E5:75_1"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">p2</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E3:75"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="cat_3.html#E2:75_1"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/><b>hence </b><font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">p</font></span>)</span> = 
 <a href="cat_3.html#K9" title="CAT_3:func.9">term</a> <font color="Maroon" title="c2">c</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E2:75"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="cat_3.html#T40" target="_self" title="CAT_3:th.40">Th40</a></i>
<br/>.= 
<font color="Maroon" title="c4">p2</font>
<b>by </b><i><a class="txt" href="cat_3.html#E2:75"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="cat_3.html#E3:75"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="cat_3.html#T40" target="_self" title="CAT_3:th.40">Th40</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:75"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="txt" href="cat_3.html#E1:75"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="cat_3.html#E3:75"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="cat_3.html#T63" target="_self" title="CAT_3:th.63">Th63</a></i>;<br/>


</div>
