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

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x2</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">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>











<b>set </b><font color="Maroon">F</font> = <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>;<br/>
<b>set </b><font color="Maroon">I</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span>;<br/>
<b>assume </b><a NAME="E1:70"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font>
 ;<br/>

<b>thus </b><a NAME="E2:70"/>
( <font color="Maroon">c</font> <a href="cat_3.html#R2" target="_self">is_a_product_wrt</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> implies <font color="Maroon">c</font> <a href="cat_3.html#R1" target="_self">is_a_product_wrt</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c</font> <a href="cat_3.html#R2" target="_self">is_a_product_wrt</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/>

<a NAME="E2:70_1"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:70_1"/>
<font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> is    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">c</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><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">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_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">b</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span>;<br/>



<b>assume </b><a NAME="E4:70_1"/><i><font color="Green">E43</font></i>: 
 <a href="cat_3.html#K4" target="_self">cods</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</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/>

<b>set </b><font color="Maroon">f</font> = <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font>;<br/>
<a NAME="E5:70_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E6:70_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="cat_3.html#K4" target="_self">cods</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_3.html#K4" target="_self">cods</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:70_1"/><b>then </b>
(  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self">Th1</a>, , </i>;<br/>
<a NAME="E8:70_1"/><b>then </b>
(  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">p1</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</font> &amp;  <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">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <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">x2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:70_1"><i><font color="Green">E42</font></i></a>, , , </i>;<br/>
<a NAME="E9:70_1"/><b>then </b>
( <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p1</font></span>)</span> &amp; <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</font></span>)</span> )
 
;<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="E11:70_1"/><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">b</font>,<font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E12:70_1"/><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">b</font>,<font color="Maroon">c</font> holds <br/>( ( <font color="Maroon">p1</font> <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="Maroon">x1</font> &amp; <font color="Maroon">p2</font> <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="Maroon">x2</font> ) iff <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E4:70_1"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E13:70_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>by </b><i/>;<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="E14:70_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 ;<br/>

<b>thus </b><a NAME="E15:70_1"/>
( ( 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> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</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">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1_1"/>
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> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</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/>

<a NAME="E2:70_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</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="Maroon">x1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</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="Maroon">x2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:70_1_1"/><b>then </b>
( <font color="Maroon">p1</font> <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="Maroon">x1</font> &amp; <font color="Maroon">p2</font> <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="Maroon">x2</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:70_1"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:70_1_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="cat_3.html#T2" target="_self">Th2</a></i>;<br/>


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

<b>assume </b><a NAME="E16:70_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">h</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="E17:70_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<a NAME="E18:70_1"/>
( <font color="Maroon">p1</font> <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="Maroon">x1</font> &amp; <font color="Maroon">p2</font> <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="Maroon">x2</font> )
 
<b>by </b><i>, <a class="ref" href="cat_3.html#T2" target="_self">Th2</a>, <a class="ref" href="cat_3.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E19:70_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</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="Maroon">x1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</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="Maroon">x2</font> &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> ) )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:70_1"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E20:70_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</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> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
 ;<br/>


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

<b>assume </b><a NAME="E3:70"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c</font> <a href="cat_3.html#R1" target="_self">is_a_product_wrt</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/>

<a NAME="E4:70"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> is    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">c</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:70"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:70"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D16" target="_self">CAT_3:def 16</a><br/>

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





<b>assume </b><b>that </b><br/><a NAME="E7:70"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p1</font></span>)</span>
 <b>and </b><br/><a NAME="E8:70"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</font></span>)</span>
 ;<br/>

<a NAME="E9:70"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</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/>
<b>then reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> as    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">d</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> <b>by </b><i/>;<br/>
 <a href="cat_3.html#K4" target="_self">cods</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> = 
<font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</font></span>)</span>
<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/>.= 
<font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>
<br/>.= 
 <a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">F'</font>
<b>by </b><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="E13:70"/><i><font color="Green">E102</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">d</font>,<font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E14:70"/><i><font color="Green">E103</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">d</font>,<font color="Maroon">c</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> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</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="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>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E15:70"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">c</font>
 <b>by </b><i/>;<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="E16:70"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">c</font>
 ;<br/>

<b>thus </b><a NAME="E17:70"/>
( <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> implies <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_3"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:70_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/><a NAME="E2:70_3_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:70_3_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> ) or ( <span class="p1">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:70_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</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> <a href="hidden.html#R1">=</a> <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#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:70_3"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="cat_3.html#T2" target="_self">Th2</a></i>;<br/>


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

<b>assume </b><a NAME="E18:70"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E19:70"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> &amp; ( 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> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</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> ) )
 
<b>by </b><i>, <a class="ref" href="cat_3.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E20:70"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</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="Maroon">x1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</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="Maroon">x2</font> )
 
;<br/>
<a NAME="E21:70"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</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> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E22:70"/>
( <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 <b>by </b><i>, </i>;<br/>


</div>
