<?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>, <font color="Maroon" title="c3">d</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="c4">p1</font>, <font color="Maroon" title="c5">p2</font>, <font color="Maroon" title="c6">q1</font>, <font color="Maroon" title="c7">q2</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>assume </b><b>that </b><br/><a NAME="E1:72"/><i><font color="Green" title="E42">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="c4">p1</font>,<font color="Maroon" title="c5">p2</font>
 <b>and </b><br/><a NAME="E2:72"/><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c3">d</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>
 <b>and </b><br/><a NAME="E3:72"/><i><font color="Green" title="E44">A3</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">q1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c7">q2</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c8">I</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c9">F</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>;<br/>
<b>set </b><font color="Maroon" title="c10">F'</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><a NAME="E1:72_1"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c4">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#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c7">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:72"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="cat_3.html#E2:72"><i><font color="Green" title="E43">A2</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="E2:72_1"/>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c2">c</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c3">d</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="ref" href="cat_3.html#T48" target="_self" title="CAT_3:th.48">Th48</a></i>;<br/><b>thus </b><a NAME="E3:72_1"/>
( <font color="Maroon" title="c2">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> &amp; <font color="Maroon" title="c3">d</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:72"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="cat_3.html#E2:72"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="cat_3.html#T59" target="_self" title="CAT_3:th.59">Th59</a></i>;<br/><b>thus </b> <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span>)</span> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">q1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c7">q2</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E3:72"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="cat_3.html#T11" target="_self" title="CAT_3:th.11">Th11</a></i>
<br/>.= 
 <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_3.html#T11" target="_self" title="CAT_3:th.11">Th11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:72"/>
<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="cat_3.html#T53" target="_self" title="CAT_3:th.53">Th53</a></i>;<br/>


</div>
