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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/>

<b>thus </b><a NAME="E1:4"/>
( <font color="Maroon" title="c1">C</font> <a href="cat_4.html#NR2" title="CAT_4:NR.2">has_finite_product</a>  implies (  ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <font color="Olive" title="b1">a</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a> &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1"/><i><font color="Green" title="E2">A1</font></i>: 
for <font color="Olive" title="b1">I</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">I</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">F'</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">I</font> st <br/>(  <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Olive" title="b4">F'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b4">F'</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_4.html#D3" target="_self" title="CAT_4:def.3">CAT_4:def 3</a><br/>

<b>reconsider </b><font color="Maroon" title="c2">F</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="funct_2.html#T55" title="FUNCT_2:th.55">FUNCT_2:55</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c4">F'</font> being    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c3">a</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <b> such that </b><br/><a NAME="E4:4_1"/>
 <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c4">F'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E5:4_1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c4">F'</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:4_1"><i><font color="Green" title="E2">A1</font></i></a></i>;<br/>
<a NAME="E6:4_1"/>
<font color="Maroon" title="c3">a</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a>
 
<b>by </b><i><a class="txt" href="cat_4.html#E5:4_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="cat_3.html#T55" title="CAT_3:th.55">CAT_3:55</a></i>;<br/>
<b>hence </b><a NAME="E7:4_1"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <font color="Olive" title="b1">a</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/>

<b>set </b><font color="Maroon" title="c7">F</font> = 0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>;<br/>
<b>consider </b><font color="Maroon" title="c8">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c9">F'</font> being    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c8">c</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span><b> such that </b><br/><a NAME="E9:4_1"/><i><font color="Green" title="E4">A3</font></i>: 
 <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c9">F'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> 0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
 <b>and </b><br/><a NAME="E10:4_1"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c9">F'</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:4_1"><i><font color="Green" title="E2">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">c</font>
;<br/>

<b>take </b><font color="Maroon" title="c10">p1</font> = <font color="Maroon" title="c9">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 0;<br/>

<b>take </b><font color="Maroon" title="c11">p2</font> = <font color="Maroon" title="c9">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1;<br/>

<a NAME="E11:4_1"/><i><font color="Green" title="E6">A5</font></i>: 
( 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E12:4_1"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="ref" href="cat_3.html#T45" title="CAT_3:th.45">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" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="ref" href="cat_3.html#T7" title="CAT_3:th.7">CAT_3:7</a></i>;<br/>
<b>hence </b><a NAME="E14:4_1"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E9:4_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="cat_4.html#E11:4_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></i>;<br/>

<a NAME="E15:4_1"/><i><font color="Green" title="E7">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">F'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:4_1"/>
( <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 
<b>by </b><i><a class="txt" href="cat_4.html#E11:4_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>;<br/>
<a NAME="E17:4_1"/><b>then </b>
( 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c9">F'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> 0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font> )
 
<b>by </b><i><a class="txt" href="cat_4.html#E15:4_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="funct_4.html#T69" title="FUNCT_4:th.69">FUNCT_4:69</a></i>;<br/>
<b>hence </b><a NAME="E18:4_1"/>
<font color="Maroon" title="c8">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E10:4_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="cat_3.html#T59" title="CAT_3:th.59">CAT_3:59</a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c2">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><b> such that </b><a NAME="E3:4"/><i><font color="Green" title="E2">A7</font></i>: 
<font color="Maroon" title="c2">a</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a>
 ;<br/>

<b>assume </b><a NAME="E4:4"/><i><font color="Green" title="E3">A8</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">I</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">I</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> ex <font color="Olive" title="b3">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">F'</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">I</font> st <br/>(  <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Olive" title="b4">F'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b4">F'</font> );<br/>
<a NAME="E5:4"/><i><font color="Green" title="E4">A9</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" title="c3">I</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">I</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>;<br/>



<b>assume </b><a NAME="E1:4_2"/><i><font color="Green" title="E5">A10</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:4_2"/><b>then </b><i><font color="Green" title="E6">A11</font></i>: 
<font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E3:4_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T55" title="FUNCT_2:th.55">FUNCT_2:55</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">F'</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">I</font> <b>by </b><i><a class="txt" href="cat_4.html#E2:4_2"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="cat_3.html#T46" title="CAT_3:th.46">CAT_3:46</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">a</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">F'</font>
;<br/>

<a NAME="E5:4_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> holds <br/><span class="p1">(<span class="default"><a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c5">F'</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_2"><i><font color="Green" title="E5">A10</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<b>hence </b><a NAME="E6:4_2"/>
 <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c5">F'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T1" title="CAT_3:th.1">CAT_3:1</a></i>;<br/>

<b>thus </b><a NAME="E7:4_2"/>
<font color="Maroon" title="c2">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c5">F'</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E3:4"><i><font color="Green" title="E2">A7</font></i></a>, <a class="txt" href="cat_4.html#E2:4_2"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="cat_3.html#T55" title="CAT_3:th.55">CAT_3:55</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:4"/><i><font color="Green" title="E5">A12</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:4_3"/><i><font color="Green" title="E6">A13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c4">I</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c5">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">I</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>;<br/>



<b>assume </b><a NAME="E2:4_3"/><i><font color="Green" title="E7">A14</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">Ix</font> = <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E5:4_3"/><i><font color="Green" title="E8">A15</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">I</font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green" title="E7">A14</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">sx</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<i><font color="Green" title="E9">A16</font></i>:  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">Ix</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c8">sx</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E5:4_3"><i><font color="Green" title="E8">A15</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon" title="c3">n</font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green" title="E7">A14</font></i></a>, <a class="ref" href="xcmplx_1.html#T26" title="XCMPLX_1:th.26">XCMPLX_1:26</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c9">G</font> = <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c11">G'</font> being    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span><b> such that </b><br/><a NAME="E10:4_3"/><i><font color="Green" title="E10">A17</font></i>: 
 <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c11">G'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">G</font>
 <b>and </b><br/><a NAME="E11:4_3"/><i><font color="Green" title="E11">A18</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c11">G'</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:4_3"><i><font color="Green" title="E6">A13</font></i></a>, <a class="txt" href="cat_4.html#E7:4_3"><i><font color="Green" title="E9">A16</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c13">p1</font>, <font color="Maroon" title="c14">p2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E13:4_3"/><i><font color="Green" title="E12">A19</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c13">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c14">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font> )
 <b>and </b><br/><a NAME="E14:4_3"/><i><font color="Green" title="E13">A20</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c13">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c14">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">x</font> )
 <b>and </b><br/><a NAME="E15:4_3"/><i><font color="Green" title="E14">A21</font></i>: 
<font color="Maroon" title="c12">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c13">p1</font>,<font color="Maroon" title="c14">p2</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E4:4"><i><font color="Green" title="E3">A8</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">F'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span>;<br/>
<a NAME="E16:4_3"/><i><font color="Green" title="E15">A22</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="cat_3.html#K1" title="CAT_3:func.1">--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E16">A23</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c4">I</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon" title="c4">I</font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green" title="E7">A14</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
;<br/>
<a NAME="E18:4_3"/><b>then </b><i><font color="Green" title="E17">A24</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E19:4_3"/><i><font color="Green" title="E18">A25</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></i>;<br/>
<a NAME="E20:4_3"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="cat_3.html#K1" title="CAT_3:func.1">--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font> )
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">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" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E22:4_3"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E19:4_3"><i><font color="Green" title="E18">A25</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">F'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">I</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="cat_4.html#E18:4_3"><i><font color="Green" title="E17">A24</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E24:4_3"/><i><font color="Green" title="E19">A26</font></i>: 
 <a href="cat_3.html#K3" title="CAT_3:func.3">doms</a> <font color="Maroon" title="c11">G'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="cat_3.html#K1" title="CAT_3:func.1">--&gt;</a> <font color="Maroon" title="c10">a</font>
 
<b>by </b><i><a class="ref" href="cat_3.html#D14" title="CAT_3:def.14">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" title="c17">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:4_3_3"/><i><font color="Green" title="E20">A27</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_3_3_1_1"/>
( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> or <font color="Maroon" title="c17">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_3_1_1_1"><b>suppose </b></a><a NAME="E1:4_3_3_1_1_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/><div class="add"><a NAME="E2:4_3_3_1_1_1"/><b>then </b><i><font color="Green" title="E21">A28</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><font color="Maroon" title="c16">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">y</font> = 
<font color="Maroon" title="c16">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green" title="E20">A27</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green" title="E15">A22</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_3_1_1_1"><i><font color="Green" title="E21">A28</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c14">p2</font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3_3_1_1_1"><i><font color="Green" title="E21">A28</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">--&gt;</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> = 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">y</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E13:4_3"><i><font color="Green" title="E12">A19</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green" title="E20">A27</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">doms</a> <font color="Maroon" title="c16">F'</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green" title="E20">A27</font></i></a>, <a class="ref" href="cat_3.html#D3" title="CAT_3:def.3">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" title="E21">A29</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font>
 ;<br/><div class="add"><a NAME="E2:4_3_3_1_1_2"/><b>then </b><i><font color="Green" title="E22">A30</font></i>: 
not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:4_3_3_1_1_2"/><i><font color="Green" title="E23">A31</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green" title="E20">A27</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_3_1_1_2"><i><font color="Green" title="E21">A29</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><font color="Maroon" title="c16">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">y</font> = 
<font color="Maroon" title="c16">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green" title="E20">A27</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green" title="E15">A22</font></i></a>, <a class="txt" href="cat_4.html#E17:4_3"><i><font color="Green" title="E16">A23</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green" title="E20">A27</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_3_1_1_2"><i><font color="Green" title="E22">A30</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_3_1_1_2"><i><font color="Green" title="E23">A31</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">doms</a> <font color="Maroon" title="c16">F'</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">y</font> = 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">y</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green" title="E20">A27</font></i></a>, <a class="ref" href="cat_3.html#D3" title="CAT_3:def.3">CAT_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_3_1_1_2"><i><font color="Green" title="E23">A31</font></i></a>, <a class="ref" href="cat_3.html#D3" title="CAT_3:def.3">CAT_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="cat_3.html#K1" title="CAT_3:func.1">--&gt;</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E13:4_3"><i><font color="Green" title="E12">A19</font></i></a>, <a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green" title="E13">A20</font></i></a>, <a class="txt" href="cat_4.html#E24:4_3"><i><font color="Green" title="E19">A26</font></i></a>, <a class="ref" href="cat_3.html#T20" title="CAT_3:th.20">CAT_3:20</a></i>
<br/>.= 
<font color="Maroon" title="c12">c</font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_3_1_1_2"><i><font color="Green" title="E23">A31</font></i></a>, <a class="ref" href="cat_3.html#T2" title="CAT_3:th.2">CAT_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">--&gt;</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green" title="E20">A27</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">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" title="CAT_3:func.3">doms</a> <font color="Maroon" title="c16">F'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">--&gt;</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_3"><i><font color="Green" title="E20">A27</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">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" title="CAT_3:func.3">doms</a> <font color="Maroon" title="c16">F'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">--&gt;</a> <font color="Maroon" title="c12">c</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">F'</font> = <font color="Maroon" title="c16">F'</font> as    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c12">c</font>,<font color="Maroon" title="c4">I</font> <b>by </b><i><a class="ref" href="cat_3.html#D14" title="CAT_3:def.14">CAT_3:def 14</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">c</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c17">F'</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:4_3_4"/><i><font color="Green" title="E20">A32</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_3_4_1_1"/>
( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_4_1_1_1"><b>suppose </b></a><a NAME="E1:4_3_4_1_1_1"/><i><font color="Green" title="E21">A33</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/><div class="add"><a NAME="E2:4_3_4_1_1_1"/><b>then </b><i><font color="Green" title="E22">A34</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><font color="Maroon" title="c17">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">y</font> = 
<font color="Maroon" title="c17">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green" title="E20">A32</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green" title="E15">A22</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_4_1_1_1"><i><font color="Green" title="E22">A34</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c14">p2</font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3_4_1_1_1"><i><font color="Green" title="E22">A34</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">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" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c17">F'</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">y</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green" title="E13">A20</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green" title="E20">A32</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_4_1_1_1"><i><font color="Green" title="E21">A33</font></i></a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">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" title="E21">A35</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font>
 ;<br/><div class="add"><a NAME="E2:4_3_4_1_1_2"/><b>then </b><i><font color="Green" title="E22">A36</font></i>: 
not <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:4_3_4_1_1_2"/><i><font color="Green" title="E23">A37</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green" title="E20">A32</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_4_1_1_2"><i><font color="Green" title="E21">A35</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><font color="Maroon" title="c17">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">y</font> = 
<font color="Maroon" title="c17">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green" title="E20">A32</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green" title="E15">A22</font></i></a>, <a class="txt" href="cat_4.html#E17:4_3"><i><font color="Green" title="E16">A23</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green" title="E20">A32</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_4_1_1_2"><i><font color="Green" title="E22">A36</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_4_1_1_2"><i><font color="Green" title="E23">A37</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c17">F'</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">y</font> = 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">y</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green" title="E20">A32</font></i></a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_4_1_1_2"><i><font color="Green" title="E23">A37</font></i></a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c9">G</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E10:4_3"><i><font color="Green" title="E10">A17</font></i></a>, <a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green" title="E13">A20</font></i></a>, <a class="txt" href="cat_4.html#E24:4_3"><i><font color="Green" title="E19">A26</font></i></a>, <a class="ref" href="cat_3.html#T20" title="CAT_3:th.20">CAT_3:20</a></i>
<br/>.= 
<font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_4_1_1_2"><i><font color="Green" title="E23">A37</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_4_1_1_2"><i><font color="Green" title="E23">A37</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_4"><i><font color="Green" title="E20">A32</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">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" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c17">F'</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">y</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E29:4_3"/><i><font color="Green" title="E20">A38</font></i>: 
 <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c17">F'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T1" title="CAT_3:th.1">CAT_3:1</a></i>;<br/>

<b>thus </b><a NAME="E30:4_3"/>
<font color="Maroon" title="c17">F'</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c12">c</font>,<font color="Maroon" title="c4">I</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D15" title="CAT_3:def.15">CAT_3:def 15</a><br/>

<b>let </b><font color="Maroon" title="c18">d</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c19">F''</font> be    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c4">I</font>;<br/>



<b>assume </b><a NAME="E31:4_3"/><i><font color="Green" title="E21">A39</font></i>: 
 <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c17">F'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c19">F''</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c20">G''</font> = <font color="Maroon" title="c19">F''</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">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" title="c21">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:4_3_5"/><i><font color="Green" title="E22">A40</font></i>: 
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><b>then </b><font color="Maroon" title="c20">G''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c21">y</font> = 
<font color="Maroon" title="c20">G''</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font>
<b>by </b><i><a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c19">F''</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_5"><i><font color="Green" title="E22">A40</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c21">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_5"><i><font color="Green" title="E22">A40</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">doms</a> <font color="Maroon" title="c20">G''</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c21">y</font> = 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c21">y</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_5"><i><font color="Green" title="E22">A40</font></i></a>, <a class="ref" href="cat_3.html#D3" title="CAT_3:def.3">CAT_3:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c18">d</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_5"><i><font color="Green" title="E22">A40</font></i></a>, <a class="ref" href="cat_3.html#T45" title="CAT_3:th.45">CAT_3:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="cat_3.html#K1" title="CAT_3:func.1">--&gt;</a> <font color="Maroon" title="c18">d</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c21">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_5"><i><font color="Green" title="E22">A40</font></i></a>, <a class="ref" href="cat_3.html#T2" title="CAT_3:th.2">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" title="CAT_3:func.3">doms</a> <font color="Maroon" title="c20">G''</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="cat_3.html#K1" title="CAT_3:func.1">--&gt;</a> <font color="Maroon" title="c18">d</font>
 
<b>by </b><i><a class="ref" href="cat_3.html#T1" title="CAT_3:th.1">CAT_3:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">G''</font> = <font color="Maroon" title="c20">G''</font> as    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <b>by </b><i><a class="ref" href="cat_3.html#D14" title="CAT_3:def.14">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" title="c22">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:4_3_6"/><i><font color="Green" title="E22">A41</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><b>then </b><i><font color="Green" title="E23">A42</font></i>: <font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">y</font> = 
<font color="Maroon" title="c19">F''</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font>
<b>by </b><i><a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c21">G''</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_6"><i><font color="Green" title="E22">A41</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c21">G''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_6"><i><font color="Green" title="E22">A41</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
;<br/><font color="Maroon" title="c9">G</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">y</font> = 
<font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_6"><i><font color="Green" title="E22">A41</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_6"><i><font color="Green" title="E22">A41</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_6"><i><font color="Green" title="E22">A41</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c11">G'</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">y</font> = 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">G''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">y</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E10:4_3"><i><font color="Green" title="E10">A17</font></i></a>, <a class="txt" href="cat_4.html#E29:4_3"><i><font color="Green" title="E20">A38</font></i></a>, <a class="txt" href="cat_4.html#E31:4_3"><i><font color="Green" title="E21">A39</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_6"><i><font color="Green" title="E22">A41</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_6"><i><font color="Green" title="E23">A42</font></i></a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c21">G''</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_6"><i><font color="Green" title="E22">A41</font></i></a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">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" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c11">G'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c21">G''</font>
 
<b>by </b><i><a class="ref" href="cat_3.html#T1" title="CAT_3:th.1">CAT_3:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">h'</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E39:4_3"/><i><font color="Green" title="E22">A43</font></i>: 
<font color="Maroon" title="c22">h'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c10">a</font>
 <b>and </b><br/><a NAME="E40:4_3"/><i><font color="Green" title="E23">A44</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c10">a</font> holds <br/>( ( for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">G''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">y</font> ) iff <font color="Maroon" title="c22">h'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E11:4_3"><i><font color="Green" title="E11">A18</font></i></a>, <a class="ref" href="cat_3.html#D15" title="CAT_3:def.15">CAT_3:def 15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c23">g</font> = <font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">x</font>;<br/>
<a NAME="E41:4_3"/><i><font color="Green" title="E24">A45</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">d</font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green" title="E7">A14</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="cat_3.html#T45" title="CAT_3:th.45">CAT_3:45</a></i>;<br/>
<a NAME="E42:4_3"/><i><font color="Green" title="E25">A46</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green" title="E26">A47</font></i>: <font color="Maroon" title="c17">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c17">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green" title="E7">A14</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green" title="E15">A22</font></i></a>, <a class="txt" href="cat_4.html#E42:4_3"><i><font color="Green" title="E25">A46</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c14">p2</font>
<b>by </b><i><a class="txt" href="cat_4.html#E42:4_3"><i><font color="Green" title="E25">A46</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<b>then </b> <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c14">p2</font> = 
<span class="p1">(<span class="default"><a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c17">F'</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green" title="E7">A14</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></i>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green" title="E7">A14</font></i></a>, <a class="txt" href="cat_4.html#E31:4_3"><i><font color="Green" title="E21">A39</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></i>
;<br/>
<a NAME="E45:4_3"/><b>then </b>
<font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c18">d</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c14">p2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E41:4_3"><i><font color="Green" title="E24">A45</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E47:4_3"/><i><font color="Green" title="E27">A48</font></i>: 
<font color="Maroon" title="c24">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font>
 <b>and </b><br/><a NAME="E48:4_3"/><i><font color="Green" title="E28">A49</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font> holds <br/>( ( <font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h'</font> &amp; <font color="Maroon" title="c14">p2</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">x</font> ) iff <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green" title="E13">A20</font></i></a>, <a class="txt" href="cat_4.html#E15:4_3"><i><font color="Green" title="E14">A21</font></i></a>, <a class="txt" href="cat_4.html#E39:4_3"><i><font color="Green" title="E22">A43</font></i></a>, <a class="ref" href="cat_3.html#D16" title="CAT_3:def.16">CAT_3:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c24">h</font>
;<br/>

<b>thus </b><a NAME="E49:4_3"/>
<font color="Maroon" title="c24">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E47:4_3"><i><font color="Green" title="E27">A48</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c25">k</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>;<br/>

<b>assume </b><a NAME="E50:4_3"/><i><font color="Green" title="E29">A50</font></i>: 
<font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font>
 ;<br/>

<b>thus </b><a NAME="E51:4_3"/>
( ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font> ) implies <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_3_9"/><i><font color="Green" title="E30">A51</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font>
 ;<br/>

<a NAME="E2:4_3_9"/><b>then </b><i><font color="Green" title="E31">A52</font></i>: 
<font color="Maroon" title="c14">p2</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3"><i><font color="Green" title="E7">A14</font></i></a>, <a class="txt" href="cat_4.html#E43:4_3"><i><font color="Green" title="E26">A47</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<a NAME="E3:4_3_9"/><i><font color="Green" title="E32">A53</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">d</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font> )
 
<b>by </b><i><a class="txt" href="cat_4.html#E50:4_3"><i><font color="Green" title="E29">A50</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>
<a NAME="E4:4_3_9"/><b>then </b>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">d</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> )
 
<b>by </b><i><a class="txt" href="cat_4.html#E13:4_3"><i><font color="Green" title="E12">A19</font></i></a>, <a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green" title="E13">A20</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/>
<a NAME="E5:4_3_9"/><b>then </b><i><font color="Green" title="E33">A54</font></i>: 
<font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c10">a</font>
 
;<br/>
<a NAME="E6:4_3_9"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">G''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:4_3_9_1"/><i><font color="Green" title="E34">A55</font></i>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:4_3_9_1"/><b>then </b><i><font color="Green" title="E35">A56</font></i>: 
( <font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<i><font color="Green" title="E36">A57</font></i>: <font color="Maroon" title="c17">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font> = 
<font color="Maroon" title="c17">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9_1"><i><font color="Green" title="E34">A55</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green" title="E15">A22</font></i></a>, <a class="txt" href="cat_4.html#E17:4_3"><i><font color="Green" title="E16">A23</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_9_1"><i><font color="Green" title="E35">A56</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9_1"><i><font color="Green" title="E34">A55</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
;<br/>
<a NAME="E4:4_3_9_1"/>
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9_1"><i><font color="Green" title="E34">A55</font></i></a>, <a class="ref" href="cat_3.html#T45" title="CAT_3:th.45">CAT_3:45</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font>
<b>by </b><i><a class="txt" href="cat_4.html#E13:4_3"><i><font color="Green" title="E12">A19</font></i></a>, <a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green" title="E13">A20</font></i></a>, <a class="txt" href="cat_4.html#E3:4_3_9"><i><font color="Green" title="E32">A53</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9_1"><i><font color="Green" title="E34">A55</font></i></a>, <a class="ref" href="cat_3.html#D7" title="CAT_3:def.7">CAT_3:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9"><i><font color="Green" title="E30">A51</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_9_1"><i><font color="Green" title="E34">A55</font></i></a>, <a class="txt" href="cat_4.html#E3:4_3_9_1"><i><font color="Green" title="E36">A57</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">F''</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9_1"><i><font color="Green" title="E34">A55</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c21">G''</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9_1"><i><font color="Green" title="E34">A55</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c21">G''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E1:4_3_9_1"><i><font color="Green" title="E34">A55</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">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" title="c13">p1</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h'</font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E40:4_3"><i><font color="Green" title="E23">A44</font></i></a>, <a class="txt" href="cat_4.html#E5:4_3_9"><i><font color="Green" title="E33">A54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:4_3_9"/>
<font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E48:4_3"><i><font color="Green" title="E28">A49</font></i></a>, <a class="txt" href="cat_4.html#E50:4_3"><i><font color="Green" title="E29">A50</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_9"><i><font color="Green" title="E31">A52</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E52:4_3"/><i><font color="Green" title="E30">A58</font></i>: 
<font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c26">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E53:4_3"/><i><font color="Green" title="E31">A59</font></i>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_10"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_3_10_1"/>
( <font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> or <font color="Maroon" title="c26">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_10_1_1"><b>suppose </b></a><a NAME="E1:4_3_10_1_1"/><i><font color="Green" title="E32">A60</font></i>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/><div class="add"><a NAME="E2:4_3_10_1_1"/><b>then </b><i><font color="Green" title="E33">A61</font></i>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><font color="Maroon" title="c17">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font> = 
<font color="Maroon" title="c17">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E53:4_3"><i><font color="Green" title="E31">A59</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green" title="E15">A22</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_10_1_1"><i><font color="Green" title="E33">A61</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c14">p2</font>
<b>by </b><i><a class="txt" href="cat_4.html#E2:4_3_10_1_1"><i><font color="Green" title="E33">A61</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">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" title="c17">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E48:4_3"><i><font color="Green" title="E28">A49</font></i></a>, <a class="txt" href="cat_4.html#E50:4_3"><i><font color="Green" title="E29">A50</font></i></a>, <a class="txt" href="cat_4.html#E52:4_3"><i><font color="Green" title="E30">A58</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_10_1_1"><i><font color="Green" title="E32">A60</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" title="E32">A62</font></i>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font>
 ;<br/><div class="add"><a NAME="E2:4_3_10_1_2"/><b>then </b><i><font color="Green" title="E33">A63</font></i>: 
not <font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:4_3_10_1_2"/><i><font color="Green" title="E34">A64</font></i>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="cat_4.html#E53:4_3"><i><font color="Green" title="E31">A59</font></i></a>, <a class="txt" href="cat_4.html#E1:4_3_10_1_2"><i><font color="Green" title="E32">A62</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E4:4_3_10_1_2"/><b>then </b><i><font color="Green" title="E35">A65</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E50:4_3"><i><font color="Green" title="E29">A50</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a>, <a class="ref" href="cat_3.html#T45" title="CAT_3:th.45">CAT_3:45</a></i>;<br/><font color="Maroon" title="c17">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font> = 
<font color="Maroon" title="c17">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E53:4_3"><i><font color="Green" title="E31">A59</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E16:4_3"><i><font color="Green" title="E15">A22</font></i></a>, <a class="txt" href="cat_4.html#E17:4_3"><i><font color="Green" title="E16">A23</font></i></a>, <a class="txt" href="cat_4.html#E53:4_3"><i><font color="Green" title="E31">A59</font></i></a>, <a class="txt" href="cat_4.html#E2:4_3_10_1_2"><i><font color="Green" title="E33">A63</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_10_1_2"><i><font color="Green" title="E34">A64</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c13">p1</font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_10_1_2"><i><font color="Green" title="E34">A64</font></i></a>, <a class="ref" href="cat_3.html#D7" title="CAT_3:def.7">CAT_3:def 7</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c17">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_4.html#E13:4_3"><i><font color="Green" title="E12">A19</font></i></a>, <a class="txt" href="cat_4.html#E14:4_3"><i><font color="Green" title="E13">A20</font></i></a>, <a class="txt" href="cat_4.html#E4:4_3_10_1_2"><i><font color="Green" title="E35">A65</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c22">h'</font>
<b>by </b><i><a class="txt" href="cat_4.html#E48:4_3"><i><font color="Green" title="E28">A49</font></i></a>, <a class="txt" href="cat_4.html#E50:4_3"><i><font color="Green" title="E29">A50</font></i></a>, <a class="txt" href="cat_4.html#E52:4_3"><i><font color="Green" title="E30">A58</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c21">G''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E39:4_3"><i><font color="Green" title="E22">A43</font></i></a>, <a class="txt" href="cat_4.html#E40:4_3"><i><font color="Green" title="E23">A44</font></i></a>, <a class="txt" href="cat_4.html#E3:4_3_10_1_2"><i><font color="Green" title="E34">A64</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c21">G''</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_10_1_2"><i><font color="Green" title="E34">A64</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c19">F''</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E3:4_3_10_1_2"><i><font color="Green" title="E34">A64</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="cat_4.html#E53:4_3"><i><font color="Green" title="E31">A59</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">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" title="c17">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F''</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">y</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:4"/><i><font color="Green" title="E6">A66</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="cat_4.html#E5:4"><i><font color="Green" title="E4">A9</font></i></a>, <a class="txt" href="cat_4.html#E6:4"><i><font color="Green" title="E5">A12</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c3">I</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_4.html#D3" target="_self" title="CAT_4:def.3">CAT_4:def 3</a><br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">I</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>;<br/>



<a NAME="E8:4"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">I</font>
 
;<br/>
<b>hence </b><a NAME="E9:4"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">F'</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font> st <br/>(  <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Olive" title="b2">F'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b2">F'</font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E7:4"><i><font color="Green" title="E6">A66</font></i></a></i>;<br/>


</div>
