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

<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/>

<b>thus </b><a NAME="E1:4"/>
( <font color="Maroon">C</font> is <a href="cat_4.html#V1" target="_self">with_finite_product</a> implies (  ex <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> st <font color="Olive">a</font> is <a href="cat_1.html#V6">terminal</a> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>  ex <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> st <br/>(  <a href="cat_1.html#K2">dom</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Olive">c</font> <a href="cat_3.html#R2">is_a_product_wrt</a> <font color="Olive">p1</font>,<font color="Olive">p2</font> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">I</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Olive">I</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>  ex <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> ex <font color="Olive">F'</font> being   <a href="cat_3.html#M1">Projections_family</a> of <font color="Olive">a</font>,<font color="Olive">I</font> st <br/>(  <a href="cat_3.html#K4">cods</a> <font color="Olive">F'</font> <a href="funct_2.html#R2">=</a> <font color="Olive">F</font> &amp; <font color="Olive">a</font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Olive">F'</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_4.html#D3" target="_self">CAT_4:def 3</a><br/>

<b>reconsider </b><font color="Maroon">F</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="funct_2.html#NM1">Function</a> of  <a href="xboole_0.html#K1">{}</a> ,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font> <b>by </b><i><a class="ref" href="funct_2.html#T55">FUNCT_2:55</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">F'</font> being    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">a</font>, <a href="xboole_0.html#K1">{}</a> <b> such that </b><br/><a NAME="E4:4_1"/>
 <a href="cat_3.html#K4">cods</a> <font color="Maroon">F'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E5:4_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Maroon">F'</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:4_1"/>
<font color="Maroon">a</font> is <a href="cat_1.html#V6">terminal</a>
 
<b>by </b><i>, <a class="ref" href="cat_3.html#T55">CAT_3:55</a></i>;<br/>
<b>hence </b><a NAME="E7:4_1"/>
 ex <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> st <font color="Olive">a</font> is <a href="cat_1.html#V6">terminal</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>, <font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/>

<b>set </b><font color="Maroon">F</font> = 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">F'</font> being    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">c</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E9:4_1"/><i><font color="Green">E36</font></i>: 
 <a href="cat_3.html#K4">cods</a> <font color="Maroon">F'</font> <a href="funct_2.html#R2">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E10:4_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c</font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Maroon">F'</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>take </b><font color="Maroon">p1</font> = <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> 0;<br/>

<b>take </b><font color="Maroon">p2</font> = <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> 1;<br/>

<a NAME="E11:4_1"/><i><font color="Green">E38</font></i>: 
( 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</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/>
<b>hence </b><a NAME="E12:4_1"/>
(  <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><a class="ref" href="cat_3.html#T45">CAT_3:45</a></i>;<br/>

<a NAME="E13:4_1"/>
( <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="cat_3.html#T7">CAT_3:7</a></i>;<br/>
<b>hence </b><a NAME="E14:4_1"/>
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , <a class="ref" href="cat_3.html#D4">CAT_3:def 4</a></i>;<br/>

<a NAME="E15:4_1"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:4_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> 1 )
 
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>;<br/>
<a NAME="E17:4_1"/><b>then </b>
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">F'</font> <a href="funct_2.html#R2">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 
<b>by </b><i>, <a class="ref" href="funct_4.html#T69">FUNCT_4:69</a></i>;<br/>
<b>hence </b><a NAME="E18:4_1"/>
<font color="Maroon">c</font> <a href="cat_3.html#R2">is_a_product_wrt</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i>, <a class="ref" href="cat_3.html#T59">CAT_3:59</a></i>;<br/>


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

<b>given </b><font color="Maroon">a</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font><b> such that </b><a NAME="E2:4"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> is <a href="cat_1.html#V6">terminal</a>
 ;<br/>

<b>assume </b><a NAME="E3:4"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>  ex <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> st <br/>(  <a href="cat_1.html#K2">dom</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Olive">c</font> <a href="cat_3.html#R2">is_a_product_wrt</a> <font color="Olive">p1</font>,<font color="Olive">p2</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">I</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Olive">I</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>  st  <a href="card_1.html#K4">card</a> <font color="Olive">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> ex <font color="Olive">F'</font> being   <a href="cat_3.html#M1">Projections_family</a> of <font color="Olive">a</font>,<font color="Olive">I</font> st <br/>(  <a href="cat_3.html#K4">cods</a> <font color="Olive">F'</font> <a href="funct_2.html#R2">=</a> <font color="Olive">F</font> &amp; <font color="Olive">a</font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Olive">F'</font> );<br/>
<a NAME="E4:4"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">I</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>;<br/>



<b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E44</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:4_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<a NAME="E3:4_2"/>
 <a href="xboole_0.html#K1">{}</a>  is   <a href="funct_2.html#NM1">Function</a> of  <a href="xboole_0.html#K1">{}</a> ,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T55">FUNCT_2:55</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F'</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">a</font>,<font color="Maroon">I</font> <b>by </b><i>, <a class="ref" href="cat_3.html#T46">CAT_3:46</a></i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>take </b>
<font color="Maroon">F'</font>
;<br/>

<a NAME="E5:4_2"/>
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"><a href="cat_3.html#K4">cods</a> <font color="Maroon">F'</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</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="card_2.html#T59">CARD_2:59</a></i>;<br/>
<b>hence </b><a NAME="E6:4_2"/>
 <a href="cat_3.html#K4">cods</a> <font color="Maroon">F'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T1">CAT_3:1</a></i>;<br/>

<b>thus </b><a NAME="E7:4_2"/>
<font color="Maroon">a</font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Maroon">F'</font>
 <b>by </b><i>, , <a class="ref" href="cat_3.html#T55">CAT_3:55</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:4"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:4_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>let </b><font color="Maroon">I</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>;<br/>



<b>assume </b><a NAME="E2:4_3"/><i><font color="Green">E50</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>
<b>reconsider </b><font color="Maroon">Ix</font> = <font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E5:4_3"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sx</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<i><font color="Green">E54</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">Ix</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">sx</font></span>)</span>
<b>by </b><i>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T26">XCMPLX_1:26</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font> <b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">G'</font> being    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">a</font>,<font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span><b> such that </b><br/><a NAME="E10:4_3"/><i><font color="Green">E57</font></i>: 
 <a href="cat_3.html#K4">cods</a> <font color="Maroon">G'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E11:4_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">a</font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Maroon">G'</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">p1</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">p2</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E13:4_3"/><i><font color="Green">E59</font></i>: 
(  <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>and </b><br/><a NAME="E14:4_3"/><i><font color="Green">E60</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</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>and </b><br/><a NAME="E15:4_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c</font> <a href="cat_3.html#R2">is_a_product_wrt</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">F'</font> = <span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">p2</font></span>)</span>;<br/>
<a NAME="E16:4_3"/><i><font color="Green">E62</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E63</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">p2</font></span>)</span></span>)</span> = 
<font color="Maroon">I</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">I</font>
<b>by </b><i>, <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/>
<a NAME="E18:4_3"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E19:4_3"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E20:4_3"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">p2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E21:4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E22:4_3"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F'</font> = <span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">p2</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E24:4_3"/><i><font color="Green">E66</font></i>: 
 <a href="cat_3.html#K3">doms</a> <font color="Maroon">G'</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="cat_3.html#D14">CAT_3:def 14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_3_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_3_3_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_3_1_1_1"><b>suppose </b></a><a NAME="E1:4_3_3_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:4_3_3_1_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
<font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">p2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">p2</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> = 
 <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">y</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3">doms</a> <font color="Maroon">F'</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D3">CAT_3:def 3</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_3_1_1_2"><b>suppose </b></a><a NAME="E1:4_3_3_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:4_3_3_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:4_3_3_1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
<font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , , , <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K3">doms</a> <font color="Maroon">F'</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_3.html#D3">CAT_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3">doms</a> <span class="p2">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D3">CAT_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, , , <a class="ref" href="cat_3.html#T20">CAT_3:20</a></i>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#T2">CAT_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:4_3_3"/>
<span class="p1">(<span class="default"><a href="cat_3.html#K3">doms</a> <font color="Maroon">F'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E26:4_3"/><b>then </b>
 <a href="cat_3.html#K3">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">--&gt;</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F'</font> as    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">c</font>,<font color="Maroon">I</font> <b>by </b><i><a class="ref" href="cat_3.html#D14">CAT_3:def 14</a></i>;<br/>
<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>take </b>
<font color="Maroon">F'</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_3_4"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_3_4_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_4_1_1_1"><b>suppose </b></a><a NAME="E1:4_3_4_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:4_3_4_1_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
<font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">p2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">p2</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E4:4_3_4_1_1_1"/>
<span class="p1">(<span class="default"><a href="cat_3.html#K4">cods</a> <font color="Maroon">F'</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="cat_3.html#D4">CAT_3:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_4_1_1_2"><b>suppose </b></a><a NAME="E1:4_3_4_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:4_3_4_1_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:4_3_4_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
<font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , , , <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K4">cods</a> <font color="Maroon">F'</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_3.html#D4">CAT_3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K4">cods</a> <span class="p2">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D4">CAT_3:def 4</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, , , <a class="ref" href="cat_3.html#T20">CAT_3:20</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:4_3_4"/>
<span class="p1">(<span class="default"><a href="cat_3.html#K4">cods</a> <font color="Maroon">F'</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E29:4_3"/><i><font color="Green">E79</font></i>: 
 <a href="cat_3.html#K4">cods</a> <font color="Maroon">F'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T1">CAT_3:1</a></i>;<br/>

<b>thus </b><a NAME="E30:4_3"/>
<font color="Maroon">F'</font> is    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">c</font>,<font color="Maroon">I</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D15">CAT_3:def 15</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_3.html#M1">Projections_family</a> of <font color="Maroon">d</font>,<font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E31:4_3"/><i><font color="Green">E82</font></i>: 
 <a href="cat_3.html#K4">cods</a> <font color="Maroon">F'</font> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K4">cods</a> <font color="Maroon">F''</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">G''</font> = <font color="Maroon">F''</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> <b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_3_5"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:4_3_5"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><font color="Maroon">G''</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
<font color="Maroon">G''</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">F''</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">F''</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K3">doms</a> <font color="Maroon">G''</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
 <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">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_3.html#D3">CAT_3:def 3</a></i>
<br/>.= 
<font color="Maroon">d</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#T45">CAT_3:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#T2">CAT_3:2</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E34:4_3"/><b>then </b>
 <a href="cat_3.html#K3">doms</a> <font color="Maroon">G''</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="cat_3.html#T1">CAT_3:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G''</font> = <font color="Maroon">G''</font> as    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">d</font>,<font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <b>by </b><i><a class="ref" href="cat_3.html#D14">CAT_3:def 14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_3_6"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:4_3_6"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then </b><i><font color="Green">E88</font></i>: <font color="Maroon">F''</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
<font color="Maroon">F''</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">G''</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">G''</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/><font color="Maroon">G</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K4">cods</a> <font color="Maroon">G'</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">G''</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="cat_3.html#D4">CAT_3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K4">cods</a> <font color="Maroon">G''</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D4">CAT_3:def 4</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E37:4_3"/><b>then </b>
 <a href="cat_3.html#K4">cods</a> <font color="Maroon">G'</font> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K4">cods</a> <font color="Maroon">G''</font>
 
<b>by </b><i><a class="ref" href="cat_3.html#T1">CAT_3:1</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="E39:4_3"/><i><font color="Green">E90</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">a</font>
 <b>and </b><br/><a NAME="E40:4_3"/><i><font color="Green">E91</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">a</font> holds <br/>( ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G''</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font> ) iff <font color="Maroon">h'</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i>, <a class="ref" href="cat_3.html#D15">CAT_3:def 15</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">F''</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>;<br/>
<a NAME="E41:4_3"/><i><font color="Green">E94</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">d</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="cat_3.html#T45">CAT_3:45</a></i>;<br/>
<a NAME="E42:4_3"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E96</font></i>: <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> = 
<font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">p2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">p2</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<b>then </b> <a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</font> = 
<span class="p1">(<span class="default"><a href="cat_3.html#K4">cods</a> <font color="Maroon">F'</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="cat_3.html#D4">CAT_3:def 4</a></i>
<br/>.= 
 <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">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="cat_3.html#D4">CAT_3:def 4</a></i>
;<br/>
<a NAME="E45:4_3"/><b>then </b>
<font color="Maroon">F''</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</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>
 
<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="E47:4_3"/><i><font color="Green">E98</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="E48:4_3"/><i><font color="Green">E99</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/>( ( <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">h'</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">x</font> ) iff <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i>, , , <a class="ref" href="cat_3.html#D16">CAT_3:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E49:4_3"/>
<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="E50:4_3"/><i><font color="Green">E100</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="E51:4_3"/>
( ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</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">y</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">y</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="4_3_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_3_9"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</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">y</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">y</font>
 ;<br/>

<a NAME="E2:4_3_9"/><b>then </b><i><font color="Green">E102</font></i>: 
<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">x</font>
 
<b>by </b><i>, , <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E3:4_3_9"/><i><font color="Green">E103</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E4:4_3_9"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">p1</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/>
<a NAME="E5:4_3_9"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">p1</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">d</font>,<font color="Maroon">a</font>
 
;<br/>
<a NAME="E6:4_3_9"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</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="finseq_4.html#K4">/.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_9_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:4_3_9_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:4_3_9_1"/><b>then </b><i><font color="Green">E106</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then </b><i><font color="Green">E107</font></i>: <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
<font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , , <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/>
<a NAME="E4:4_3_9_1"/>
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="cat_3.html#T45">CAT_3:45</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</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">G'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">p1</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">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</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#D7">CAT_3:def 7</a></i>
<br/>.= 
<font color="Maroon">F''</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="cat_4.html#T1" target="_self">Th1</a>, , </i>
<br/>.= 
<font color="Maroon">F''</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">G''</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">G''</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E7:4_3_9"/><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">h'</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:4_3_9"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , </i>;<br/>


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

<b>assume </b><a NAME="E52:4_3"/><i><font color="Green">E108</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">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E53:4_3"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_10"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_3_10_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_10_1_1"><b>suppose </b></a><a NAME="E1:4_3_10_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:4_3_10_1_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
<font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">p2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">p2</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E4:4_3_10_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</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">y</font>
 <b>by </b><i>, , <a class="ref" href="cat_4.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_10_1_2"><b>suppose </b></a><a NAME="E1:4_3_10_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:4_3_10_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:4_3_10_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><a NAME="E4:4_3_10_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">k</font> <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"><font color="Maroon">G'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</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#T18">CAT_1:18</a>, <a class="ref" href="cat_3.html#T45">CAT_3:45</a></i>;<br/><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
<font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , , , <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="cat_3.html#K7">*</a> <font color="Maroon">p1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">p1</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D7">CAT_3:def 7</a></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">y</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">h'</font>
<b>by </b><i>, , <a class="ref" href="cat_4.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">G''</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">G''</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">F''</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">F''</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E55:4_3"/>
<span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</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">y</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:4"/><i><font color="Green">E116</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>let </b><font color="Maroon">I</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_4.html#D3" target="_self">CAT_4:def 3</a><br/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>;<br/>



<a NAME="E7:4"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">I</font>
 
;<br/>
<b>hence </b><a NAME="E8:4"/>
 ex <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> ex <font color="Olive">F'</font> being   <a href="cat_3.html#M1">Projections_family</a> of <font color="Olive">a</font>,<font color="Maroon">I</font> st <br/>(  <a href="cat_3.html#K4">cods</a> <font color="Olive">F'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font> &amp; <font color="Olive">a</font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Olive">F'</font> )
 <b>by </b><i/>;<br/>


</div>
