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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/>

<b>thus </b><a NAME="E1:4"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="cat_4.html#NR2" target="_self">has_finite_product</a>  implies ( ex <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="cat_1.html#V6">terminal</a> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="cat_3.html#R2">is_a_product_wrt</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></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">E2</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <a href="cat_3.html#M1">Projections_family</a> of <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> st <br/>(  <a href="cat_3.html#K4">cods</a> <font color="Olive">b<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Olive">b<sub>4</sub></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">c<sub>2</sub></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<sub>1</sub></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">c<sub>3</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>4</sub></font> being    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:4_1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:4_1"><i><font color="Green">E2</font></i></a></i>;<br/>
<a NAME="E6:4_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="cat_1.html#V6">terminal</a>
 
<b>by </b><i><a class="txt" href="cat_4.html#E5:4_1"><i><font color="Green">E3</font></i></a>, <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">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="cat_1.html#V6">terminal</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">c<sub>8</sub></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">E4</font></i>: 
 <a href="cat_3.html#K4">cods</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R2">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E10:4_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:4_1"><i><font color="Green">E2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> 0;<br/>

<b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> 1;<br/>

<a NAME="E11:4_1"/><i><font color="Green">E6</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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E9:4_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="cat_4.html#E11:4_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cat_3.html#D4">CAT_3:def 4</a></i>;<br/>

<a NAME="E15:4_1"/><i><font color="Green">E7</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1 )
 
<b>by </b><i><a class="txt" href="cat_4.html#E11:4_1"><i><font color="Green">E6</font></i></a>, <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">c<sub>9</sub></font> <a href="funct_2.html#R2">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_4.html#E15:4_1"><i><font color="Green">E7</font></i></a>, <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<sub>8</sub></font> <a href="cat_3.html#R2">is_a_product_wrt</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E10:4_1"><i><font color="Green">E5</font></i></a>, <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">c<sub>2</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:4"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="cat_1.html#V6">terminal</a>
 ;<br/>

<b>assume </b><a NAME="E3:4"/><i><font color="Green">E3</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="cat_3.html#R2">is_a_product_wrt</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <a href="cat_3.html#M1">Projections_family</a> of <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> st <br/>(  <a href="cat_3.html#K4">cods</a> <font color="Olive">b<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Olive">b<sub>4</sub></font> );<br/>
<a NAME="E4:4"/><i><font color="Green">E4</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">c<sub>3</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E5</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:4_2"/><b>then </b><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></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<sub>1</sub></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">c<sub>5</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="cat_4.html#E2:4_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cat_3.html#T46">CAT_3:46</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<a NAME="E5:4_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><span class="p1">(<span class="default"><a href="cat_3.html#K4">cods</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_2"><i><font color="Green">E5</font></i></a>, <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">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E2:4"><i><font color="Green">E2</font></i></a>, <a class="txt" href="cat_4.html#E2:4_2"><i><font color="Green">E6</font></i></a>, <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">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:4_3"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E2:4_3"/><i><font color="Green">E7</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">E8</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green">E7</font></i></a>, <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">c<sub>8</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">E9</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E5:4_3"><i><font color="Green">E8</font></i></a>, <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">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xcmplx_1.html#T26">XCMPLX_1:26</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>10</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span><b> such that </b><br/><a NAME="E10:4_3"/><i><font color="Green">E10</font></i>: 
 <a href="cat_3.html#K4">cods</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E11:4_3"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:4_3"><i><font color="Green">E6</font></i></a>, <a class="txt" href="cat_4.html#E7:4_3"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:4_3"/><i><font color="Green">E12</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E14:4_3"/><i><font color="Green">E13</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E15:4_3"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="cat_3.html#R2">is_a_product_wrt</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E3:4"><i><font color="Green">E3</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>;<br/>
<a NAME="E16:4_3"/><i><font color="Green">E15</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">E16</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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green">E7</font></i></a>, <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">E17</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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">E18</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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">c<sub>14</sub></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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">c<sub>14</sub></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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">c<sub>14</sub></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<sub>1</sub></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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">c<sub>14</sub></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<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E19:4_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="cat_4.html#E18:4_3"><i><font color="Green">E17</font></i></a>, <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">E19</font></i>: 
 <a href="cat_3.html#K3">doms</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c<sub>10</sub></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">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_3_3"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></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">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:4_3_3_1_1_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green">E20</font></i></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">c<sub>6</sub></font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_3_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3_3_1_1_1"><i><font color="Green">E21</font></i></a>, <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">c<sub>4</sub></font> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> = 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E13:4_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green">E20</font></i></a>, <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">c<sub>16</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green">E20</font></i></a>, <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">E21</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:4_3_3_1_1_2"/><b>then </b><i><font color="Green">E22</font></i>: 
not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">E23</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_3_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green">E20</font></i></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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cat_4.html#E17:4_3"><i><font color="Green">E16</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_3_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_3_1_1_2"><i><font color="Green">E23</font></i></a>, <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">c<sub>16</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font> = 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green">E20</font></i></a>, <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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_3_1_1_2"><i><font color="Green">E23</font></i></a>, <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">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E13:4_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_4.html#E24:4_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="cat_3.html#T20">CAT_3:20</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_3_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="cat_3.html#T2">CAT_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green">E20</font></i></a>, <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">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green">E20</font></i></a>, <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">c<sub>16</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c<sub>12</sub></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">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>4</sub></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<sub>12</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>17</sub></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">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_3_4"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></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">E21</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:4_3_4_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green">E20</font></i></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">c<sub>6</sub></font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_4_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3_4_1_1_1"><i><font color="Green">E22</font></i></a>, <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">c<sub>17</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_4_1_1_1"><i><font color="Green">E21</font></i></a>, <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">E21</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:4_3_4_1_1_2"/><b>then </b><i><font color="Green">E22</font></i>: 
not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">E23</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_4_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green">E20</font></i></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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cat_4.html#E17:4_3"><i><font color="Green">E16</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_4_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_4_1_1_2"><i><font color="Green">E23</font></i></a>, <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">c<sub>17</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font> = 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green">E20</font></i></a>, <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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_4_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="cat_3.html#D4">CAT_3:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E10:4_3"><i><font color="Green">E10</font></i></a>, <a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_4.html#E24:4_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="cat_3.html#T20">CAT_3:20</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_4_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_4_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green">E20</font></i></a>, <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">c<sub>17</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E29:4_3"/><i><font color="Green">E20</font></i>: 
 <a href="cat_3.html#K4">cods</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></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">c<sub>17</sub></font> is    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>18</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>



<b>assume </b><a NAME="E31:4_3"/><i><font color="Green">E21</font></i>: 
 <a href="cat_3.html#K4">cods</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K4">cods</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_3_5"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:4_3_5"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font> = 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_5"><i><font color="Green">E22</font></i></a>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_5"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3_5"><i><font color="Green">E23</font></i></a>, <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">c<sub>20</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font> = 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_5"><i><font color="Green">E22</font></i></a>, <a class="ref" href="cat_3.html#D3">CAT_3:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3_5"><i><font color="Green">E23</font></i></a>, <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">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_5"><i><font color="Green">E22</font></i></a>, <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">c<sub>20</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="cat_3.html#K1">--&gt;</a> <font color="Maroon">c<sub>18</sub></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">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as    <a href="cat_3.html#M1">Projections_family</a> of <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>22</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_3_6"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:4_3_6"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then </b><i><font color="Green">E24</font></i>: <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_6"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_6"><i><font color="Green">E22</font></i></a>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/><font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_6"><i><font color="Green">E22</font></i></a>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_6"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3_6"><i><font color="Green">E23</font></i></a>, <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">c<sub>11</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>22</sub></font> = 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E10:4_3"><i><font color="Green">E10</font></i></a>, <a class="txt" href="cat_4.html#E29:4_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cat_4.html#E31:4_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_6"><i><font color="Green">E23</font></i></a>, <a class="txt" href="cat_4.html#E3:4_3_6"><i><font color="Green">E24</font></i></a>, <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">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_6"><i><font color="Green">E22</font></i></a>, <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">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K4">cods</a> <font color="Maroon">c<sub>21</sub></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">c<sub>22</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E39:4_3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E40:4_3"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>10</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> ) iff <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E11:4_3"><i><font color="Green">E11</font></i></a>, <a class="ref" href="cat_3.html#D15">CAT_3:def 15</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E41:4_3"/><i><font color="Green">E24</font></i>: 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green">E7</font></i></a>, <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">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">E26</font></i>: <font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green">E7</font></i></a>, <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">c<sub>6</sub></font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cat_4.html#E42:4_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E42:4_3"><i><font color="Green">E25</font></i></a>, <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">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><a href="cat_3.html#K4">cods</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green">E7</font></i></a>, <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">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green">E7</font></i></a>, <a class="txt" href="cat_4.html#E31:4_3"><i><font color="Green">E21</font></i></a>, <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">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E41:4_3"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E47:4_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E48:4_3"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>12</sub></font> holds <br/>( ( <font color="Maroon">c<sub>13</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font> ) iff <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_4.html#E15:4_3"><i><font color="Green">E14</font></i></a>, <a class="txt" href="cat_4.html#E39:4_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="cat_3.html#D16">CAT_3:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>24</sub></font>
;<br/>

<b>thus </b><a NAME="E49:4_3"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E47:4_3"><i><font color="Green">E27</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>25</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E50:4_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<b>thus </b><a NAME="E51:4_3"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></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">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<a NAME="E2:4_3_9"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green">E7</font></i></a>, <a class="txt" href="cat_4.html#E43:4_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E3:4_3_9"/><i><font color="Green">E32</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_4.html#E50:4_3"><i><font color="Green">E29</font></i></a>, <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">c<sub>13</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_4.html#E13:4_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green">E13</font></i></a>, <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">E33</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
;<br/>
<a NAME="E6:4_3_9"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></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">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:4_3_9_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:4_3_9_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">E36</font></i>: <font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cat_4.html#E17:4_3"><i><font color="Green">E16</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_9_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9_1"><i><font color="Green">E34</font></i></a>, <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">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9_1"><i><font color="Green">E34</font></i></a>, <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">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E13:4_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_4.html#E3:4_3_9"><i><font color="Green">E32</font></i></a>, <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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="cat_3.html#D7">CAT_3:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9"><i><font color="Green">E30</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_9_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="cat_4.html#E3:4_3_9_1"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3_9_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9_1"><i><font color="Green">E34</font></i></a>, <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">c<sub>13</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E40:4_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="cat_4.html#E5:4_3_9"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:4_3_9"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E48:4_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="cat_4.html#E50:4_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_9"><i><font color="Green">E31</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E52:4_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E53:4_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></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">E32</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:4_3_10_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E53:4_3"><i><font color="Green">E31</font></i></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">c<sub>6</sub></font> <a href="cat_3.html#K2">.--&gt;</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_10_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3_10_1_1"><i><font color="Green">E33</font></i></a>, <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">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E48:4_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="cat_4.html#E50:4_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="cat_4.html#E52:4_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_10_1_1"><i><font color="Green">E32</font></i></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">E32</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:4_3_10_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
not <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">E34</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="cat_4.html#E53:4_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_10_1_2"><i><font color="Green">E32</font></i></a>, <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">E35</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E50:4_3"><i><font color="Green">E29</font></i></a>, <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">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E53:4_3"><i><font color="Green">E31</font></i></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">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cat_4.html#E17:4_3"><i><font color="Green">E16</font></i></a>, <a class="txt" href="cat_4.html#E53:4_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_10_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_3.html#K7">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_10_1_2"><i><font color="Green">E34</font></i></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">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_10_1_2"><i><font color="Green">E34</font></i></a>, <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">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E13:4_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_4.html#E4:4_3_10_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E48:4_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="cat_4.html#E50:4_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="cat_4.html#E52:4_3"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E39:4_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="cat_4.html#E40:4_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="cat_4.html#E3:4_3_10_1_2"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_10_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_10_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="cat_4.html#E53:4_3"><i><font color="Green">E31</font></i></a>, <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">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:4"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="cat_4.html#E4:4"><i><font color="Green">E4</font></i></a>, <a class="txt" href="cat_4.html#E5:4"><i><font color="Green">E5</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E7:4"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>hence </b><a NAME="E8:4"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="cat_3.html#M1">Projections_family</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <br/>(  <a href="cat_3.html#K4">cods</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="cat_3.html#R1">is_a_product_wrt</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E6:4"><i><font color="Green">E6</font></i></a></i>;<br/>


</div>
