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

<span class="kw">let </span><font color="Maroon" title="c1">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being  <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> holds <br/><span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font> be   <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> implies <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E2:59"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">f</font> = <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f0</font> =  <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">Z</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span> ;<br/>
<a NAME="E3:59"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#D5" target="_self" title="AFINSQ_2:def.5">Def5</a></span>;<br/>
<a NAME="E4:59"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th31</a></span>;<br/>
<a NAME="E5:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E6:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T58" title="AFINSQ_1:th.58">AFINSQ_1:58</a></span>;<br/>
<a NAME="E7:59"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span>  or <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:59_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">v0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> st <br/>( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v0</font> &amp;  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:59_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">Z0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</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="E9:59"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
 <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></span>;<br/>
<a NAME="E10:59"/>
 <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> or <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_2"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_2"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:59_2"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">y0</font> = <font color="Maroon" title="c8">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E1:59_2"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="ref" href="afinsq_2.html#D5" target="_self" title="AFINSQ_2:def.5">Def5</a></span>;<br/>
<a NAME="E6:59_2"/>
 ex <font color="Olive" title="b1">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c10">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k2</font> &amp; <font color="Olive" title="b1">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E3:59_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:59_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">nx</font> = <font color="Maroon" title="c9">x</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a></span>;<br/>

</div><span class="kw">end;</span></div>
<a NAME="E11:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A11</font></span>: 
 <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E12:59"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E42">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59_3"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E2:59_3"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
not <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">consider </span><font color="Maroon" title="c8">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:59_3"/><span class="lab"><font color="Green" title="E45">A16</font></span>: 
<font color="Maroon" title="c8">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span> 
 <span class="kw">and </span><br/><a NAME="E5:59_3"/><span class="lab"><font color="Green" title="E46">A17</font></span>: 
not <font color="Maroon" title="c8">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_3"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">v10</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font><span class="kw"> such that </span><br/><a NAME="E7:59_3"/><span class="lab"><font color="Green" title="E47">A18</font></span>: 
<font color="Maroon" title="c8">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v10</font>
 <span class="kw">and </span><br/><a NAME="E8:59_3"/><span class="lab"><font color="Green" title="E48">A19</font></span>: 
 ex <font color="Olive" title="b1">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c9">v10</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k2</font> &amp; <font color="Olive" title="b1">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_3"><span class="lab"><font color="Green" title="E45">A16</font></span></a></span>;<br/><a NAME="E9:59_3"/><span class="lab"><font color="Green" title="E49">A20</font></span>: 
<font color="Maroon" title="c9">v10</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E5:59_3"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="txt" href="#E7:59_3"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">nv10</font> = <font color="Maroon" title="c9">v10</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><span class="kw">consider </span><font color="Maroon" title="c11">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E12:59_3"/><span class="lab"><font color="Green" title="E50">A21</font></span>: 
<font color="Maroon" title="c11">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E13:59_3"/><span class="lab"><font color="Green" title="E51">A22</font></span>: 
not <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_3"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E14:59_3"/>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:59_3"/><span class="lab"><font color="Green" title="E52">A23</font></span>: 
<font color="Maroon" title="c12">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:59_3"/><span class="lab"><font color="Green" title="E53">A24</font></span>: 
<font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E12:59_3"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">x20</font> = <font color="Maroon" title="c12">x2</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:59_3"><span class="lab"><font color="Green" title="E52">A23</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:59_3_1"/>
<font color="Maroon" title="c13">x20</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:59_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A25</font></span>: 
<font color="Maroon" title="c13">x20</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E3:59_3_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E4:59_3_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th31</a></span>;<br/><a NAME="E5:59_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x20</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x20</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_3_1"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="afinsq_1.html#T57" title="AFINSQ_1:th.57">AFINSQ_1:57</a></span>;<br/><span class="kw">hence </span><a NAME="E6:59_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E13:59_3"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="txt" href="#E17:59_3"><span class="lab"><font color="Green" title="E53">A24</font></span></a>, <a class="txt" href="#E2:59_3_1"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E20:59_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A26</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">x20</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="afinsq_1.html#T58" title="AFINSQ_1:th.58">AFINSQ_1:58</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">k20</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E22:59_3"/><span class="lab"><font color="Green" title="E55">A27</font></span>: 
<font color="Maroon" title="c9">v10</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">k20</font>
 <span class="kw">and </span><br/><a NAME="E23:59_3"/><span class="lab"><font color="Green" title="E56">A28</font></span>: 
<font color="Maroon" title="c14">k20</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_3"><span class="lab"><font color="Green" title="E48">A19</font></span></a></span>;<br/><a NAME="E24:59_3"/><span class="lab"><font color="Green" title="E57">A29</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">k20</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">k20</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E23:59_3"><span class="lab"><font color="Green" title="E56">A28</font></span></a>, <a class="ref" href="afinsq_1.html#T57" title="AFINSQ_1:th.57">AFINSQ_1:57</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">nv2</font> = <font color="Maroon" title="c11">v2</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:59_3"><span class="lab"><font color="Green" title="E53">A24</font></span></a></span>;<br/><a NAME="E26:59_3"/>
<font color="Maroon" title="c14">k20</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E23:59_3"><span class="lab"><font color="Green" title="E56">A28</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E27:59_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A30</font></span>: 
<font color="Maroon" title="c14">k20</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">x20</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:59_3"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E28:59_3"/>
<font color="Maroon" title="c13">x20</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:59_3"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E29:59_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">nv10</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">nv2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_3"><span class="lab"><font color="Green" title="E55">A27</font></span></a>, <a class="txt" href="#E17:59_3"><span class="lab"><font color="Green" title="E53">A24</font></span></a>, <a class="txt" href="#E27:59_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="txt" href="#E24:59_3"><span class="lab"><font color="Green" title="E57">A29</font></span></a>, <a class="ref" href="afinsq_2.html#D5" target="_self" title="AFINSQ_2:def.5">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E30:59_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E12:59_3"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="txt" href="#E9:59_3"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="ref" href="afinsq_2.html#D6" target="_self" title="AFINSQ_2:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E14:59"/>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></span>;<br/>
<a NAME="E15:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A31</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T60" title="CARD_1:th.60">CARD_1:60</a></span>;<br/>
<a NAME="E16:59"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></span>;<br/>
<a NAME="E17:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A32</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E15:59"><span class="lab"><font color="Green" title="E43">A31</font></span></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></span>;<br/>
<a NAME="E18:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A33</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> :  ex <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Olive" title="b2">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="afinsq_1.html#T58" title="AFINSQ_1:th.58">AFINSQ_1:58</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E46">A34</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:59_4_1"/>
( <font color="Maroon" title="c7">Z0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> or <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z0</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E42">A13</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_4_1_1"><span class="kw">case </span></a><a NAME="E1:59_4_1_1"/>
<font color="Maroon" title="c7">Z0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">Z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:59_4_1_1"/>
<font color="Maroon" title="c7">Z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E17:59"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="ref" href="afinsq_1.html#T58" title="AFINSQ_1:th.58">AFINSQ_1:58</a>, <a class="txt" href="afinsq_2.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_4_1_2"><span class="kw">case </span></a><a NAME="E1:59_4_1_2"/>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">Z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:59_4_1_2"/>
<font color="Maroon" title="c7">Z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="afinsq_2.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E20:59"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="afinsq_1.html#T57" title="AFINSQ_1:th.57">AFINSQ_1:57</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:59"/>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E19:59"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
