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

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">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="c1">X</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c2">Y</font> iff  <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <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> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">p</font> =  <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">q</font> =  <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">r</font> =  <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span>;<br/>
<span class="kw">thus </span><a NAME="E1:66"/>
( <font color="Maroon" title="c1">X</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c2">Y</font> implies  <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <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> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <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> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> implies <font color="Maroon" title="c1">X</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c2">Y</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <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> <font color="Maroon" title="c1">X</font></span>)</span> 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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1 <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> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1 );<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">X1</font> = <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c7">Y1</font> = <font color="Maroon" title="c2">Y</font> 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/>
<span class="kw">assume </span><a NAME="E2:66_1"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <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> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/>

<a NAME="E3:66_1"/>
<font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font>;<br/>
<a NAME="E2:66_1_1"/>
<font color="Maroon" title="c1">X</font> <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<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="E3:66_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:66_1_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
not <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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="E5:66_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66_1_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/>
<a NAME="E7:66_1_1"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:66_1_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E8:66_1_1"/>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:66_1_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:66_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E7:66_1_1"><span class="lab"><font color="Green" title="E43">A4</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="E4:66_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A5</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="lab"><font color="Green" title="E42">A6</font></span>:  <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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">Y1</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/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">X1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">Y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:66_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></span>
<br/>.= 
<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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">Y1</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/>.= 
<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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">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/>
<div><span class="lab"><font color="Green" title="E43">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">k</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:66_1_3"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</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">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]</span><br/><span class="kw">thus </span><a NAME="E2:66_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c9">m</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">n</font> = <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:66_1_3_1"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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> <font color="Maroon" title="c1">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"> <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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <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> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span><br/>

<a NAME="E2:66_1_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<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/>
<a NAME="E3:66_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A10</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</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="E4:66_1_3_1"/>
 <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> <font color="Maroon" title="c1">X</font></span>)</span> <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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E5:66_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A11</font></span>: 
 <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> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></span>;<br/>
<a NAME="E6:66_1_3_1"/><span class="kw">then </span>
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_3_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E7:66_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A12</font></span>: 
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">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/>
<span class="kw">assume </span><a NAME="E8:66_1_3_1"/><span class="lab"><font color="Green" title="E49">A13</font></span>: 
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_3_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_3_1_1_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="E1:66_1_3_1_1_1"/>
( <font color="Maroon" title="c8">k</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> <font color="Maroon" title="c1">X</font></span>)</span> or  not <font color="Maroon" title="c8">k</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> <font color="Maroon" title="c1">X</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:66_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A14</font></span>: 
<font color="Maroon" title="c8">k</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> <font color="Maroon" title="c1">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/><div class="add"><span class="kw">set </span><font color="Maroon" title="c11">m1</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font>;<br/><span class="kw">set </span><font color="Maroon" title="c12">n1</font> = <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_3_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_3_1_1_1_1_1_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="E1:66_1_3_1_1_1_1_1_1"/>
( <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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> or <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:66_1_3_1"><span class="lab"><font color="Green" title="E49">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_3_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:66_1_3_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A15</font></span>: 
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</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/><div class="add"><a NAME="E2:66_1_3_1_1_1_1_1_1_1"/><span class="kw">then </span>
not <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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E3:66_1_3_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="ref" href="afinsq_2.html#D6" target="_self" title="AFINSQ_2:def.6">Def6</a></span>;<br/><a NAME="E3:66_1_3_1_1_1_1_1_1_1"/><span class="kw">then </span>
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:66_1_3_1"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E4:66_1_3_1_1_1_1_1_1_1"/><span class="kw">then </span>
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:66_1_3_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A16</font></span>: 
<font color="Maroon" title="c13">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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:66_1_3_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E53">A17</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <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="c14">x</font> = <font color="Maroon" title="c13">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A16</font></span></a></span>;<br/><a NAME="E9:66_1_3_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c14">x</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> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A16</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E10:66_1_3_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A18</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="txt" href="#E7:66_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="ref" href="afinsq_2.html#T44" target="_self" title="AFINSQ_2:th.44">Th44</a></span>;<br/><a NAME="E11:66_1_3_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E55">A19</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E12:66_1_3_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_3_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E5:66_1_3_1"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E13:66_1_3_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A20</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_3"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E1:66_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="txt" href="#E11:66_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A19</font></span></a>, <a class="ref" href="afinsq_2.html#D5" target="_self" title="AFINSQ_2:def.5">Def5</a></span>;<br/><a NAME="E14:66_1_3_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c8">k</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> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E15:66_1_3_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:66_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="txt" href="#E13:66_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="ref" href="afinsq_2.html#T44" target="_self" title="AFINSQ_2:th.44">Th44</a></span>;<br/><span class="kw">hence </span><a NAME="E16:66_1_3_1_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A18</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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="66_1_3_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:66_1_3_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A21</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</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/><div class="add"><a NAME="E2:66_1_3_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66_1_3_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E3:66_1_3_1_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:66_1_3_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A22</font></span>: 
<font color="Maroon" title="c13">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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:66_1_3_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A23</font></span>: 
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x</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> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <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="c14">x</font> = <font color="Maroon" title="c13">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1_3_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A22</font></span></a></span>;<br/><a NAME="E8:66_1_3_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c14">x</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1_3_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E9:66_1_3_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A24</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_3_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E51">A21</font></span></a>, <a class="txt" href="#E6:66_1_3_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="ref" href="afinsq_2.html#T44" target="_self" title="AFINSQ_2:th.44">Th44</a></span>;<br/><a NAME="E10:66_1_3_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E55">A25</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E11:66_1_3_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_3_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E12:66_1_3_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A26</font></span>: 
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_3"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E1:66_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="txt" href="#E10:66_1_3_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="ref" href="afinsq_2.html#D5" target="_self" title="AFINSQ_2:def.5">Def5</a></span>;<br/><a NAME="E13:66_1_3_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">k</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1_3_1"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="txt" href="#E1:66_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E14:66_1_3_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66_1_3_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="txt" href="#E12:66_1_3_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="ref" href="afinsq_2.html#T44" target="_self" title="AFINSQ_2:th.44">Th44</a></span>;<br/><span class="kw">hence </span><a NAME="E15:66_1_3_1_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:66_1_3_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E54">A24</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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><span class="kw">hence </span><a NAME="E3:66_1_3_1_1_1_1"/>
contradiction
 ; <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="66_1_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:66_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A27</font></span>: 
not <font color="Maroon" title="c8">k</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> <font color="Maroon" title="c1">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/><div class="add"><a NAME="E2:66_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A28</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E3:66_1_3_1_1_1_2"/>
 <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> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:66_1_3_1_1_1_2"/><span class="kw">then </span>
 <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> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E51">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:66_1_3_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_3_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</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>
<span class="kw">hence </span><a NAME="E10:66_1_3_1"/>
contradiction
 ; <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>
<a NAME="E7:66_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> <font color="Maroon" title="c1">X</font></span>)</span> implies <font color="Maroon" title="c6">X1</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="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th31</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></span>;<br/>
<a NAME="E8:66_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A29</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="afinsq_2.html#T45" target="_self" title="AFINSQ_2:th.45">Th45</a></span>;<br/>
<a NAME="E9:66_1"/><span class="lab"><font color="Green" title="E45">A30</font></span>: 
for <font color="Olive" title="b1">k</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">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E8:66_1"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="txt" href="#E6:66_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <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> <font color="Maroon" title="c2">Y</font></span>)</span> 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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> $1</span>)</span> <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> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1 );<br/>
<div><span class="lab"><font color="Green" title="E46">A31</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">k</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"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:66_1_4"/><span class="lab"><font color="Green" title="E47">A32</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</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">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]</span><br/><span class="kw">thus </span><a NAME="E2:66_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c9">n</font> = <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">a</font> = <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">m</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:66_1_4_1"/><span class="lab"><font color="Green" title="E48">A33</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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> <font color="Maroon" title="c2">Y</font></span>)</span>
 ; <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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <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> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span><br/>

<a NAME="E2:66_1_4_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<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/>
<a NAME="E3:66_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A34</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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="E4:66_1_4_1"/>
<font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_4_1"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E5:66_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A35</font></span>: 
<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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E6:66_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A36</font></span>: 
<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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<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="E7:66_1_4_1"/><span class="kw">then </span>
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<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/>
<a NAME="E8:66_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A37</font></span>: 
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">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/>
<div><span class="lab"><font color="Green" title="E53">A38</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:66_1_4_1_1"/><span class="lab"><font color="Green" title="E54">A39</font></span>: 
 <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> <font color="Maroon" title="c1">X</font></span>)</span> <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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><span class="kw">assume </span><a NAME="E2:66_1_4_1_1"/>
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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="E3:66_1_4_1_1"/><span class="kw">then </span>
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:66_1_4_1_1"/><span class="lab"><font color="Green" title="E55">A40</font></span>: 
<font color="Maroon" title="c12">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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:66_1_4_1_1"/><span class="lab"><font color="Green" title="E56">A41</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">x</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <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="c13">x</font> = <font color="Maroon" title="c12">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1_4_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a></span>;<br/><a NAME="E8:66_1_4_1_1"/>
<font color="Maroon" title="c13">x</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> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1_4_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E9:66_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_4_1_1"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:66_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A42</font></span>: 
<font color="Maroon" title="c13">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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <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="E11:66_1_4_1_1"/>
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:66_1"><span class="lab"><font color="Green" title="E45">A30</font></span></a>, <a class="txt" href="#E5:66_1_4_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E6:66_1_4_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a></span>;<br/><a NAME="E12:66_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66_1_4_1"><span class="lab"><font color="Green" title="E51">A36</font></span></a>, <a class="txt" href="#E10:66_1_4_1_1"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/><a NAME="E13:66_1_4_1_1"/><span class="kw">then </span>
<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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1_4_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><span class="kw">hence </span><a NAME="E14:66_1_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</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">assume </span><a NAME="E10:66_1_4_1"/><span class="lab"><font color="Green" title="E54">A43</font></span>: 
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4_1_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_1_2_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="E1:66_1_4_1_2_1"/>
( <font color="Maroon" title="c8">k</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> <font color="Maroon" title="c2">Y</font></span>)</span> or  not <font color="Maroon" title="c8">k</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> <font color="Maroon" title="c2">Y</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:66_1_4_1_2_1_1"/><span class="lab"><font color="Green" title="E55">A44</font></span>: 
<font color="Maroon" title="c8">k</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> <font color="Maroon" title="c2">Y</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/><div class="add"><span class="kw">set </span><font color="Maroon" title="c12">m1</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c8">k</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">n1</font> = <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font>;<br/><a NAME="E2:66_1_4_1_2_1_1"/><span class="lab"><font color="Green" title="E56">A45</font></span>: 
<font color="Maroon" title="c8">k</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> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_1_2_1_1_1_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="E1:66_1_4_1_2_1_1_1_1"/>
( <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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> or <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66_1_4_1"><span class="lab"><font color="Green" title="E54">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_1_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:66_1_4_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E57">A46</font></span>: 
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</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/><div class="add"><a NAME="E2:66_1_4_1_2_1_1_1_1_1"/>
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:66_1_4_1"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="txt" href="#E9:66_1_4_1"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E3:66_1_4_1_2_1_1_1_1_1"/><span class="kw">then </span>
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:66_1_4_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E58">A47</font></span>: 
<font color="Maroon" title="c14">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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:66_1_4_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A48</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <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="c15">x</font> = <font color="Maroon" title="c14">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1_4_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A47</font></span></a></span>;<br/><a NAME="E8:66_1_4_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c15">x</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> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1_4_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E9:66_1_4_1_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A49</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_4_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E57">A46</font></span></a>, <a class="txt" href="#E6:66_1_4_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A48</font></span></a>, <a class="ref" href="afinsq_2.html#T44" target="_self" title="AFINSQ_2:th.44">Th44</a></span>;<br/><a NAME="E10:66_1_4_1_2_1_1_1_1_1"/>
<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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E11:66_1_4_1_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A50</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_4"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="txt" href="#E5:66_1_4_1"><span class="lab"><font color="Green" title="E50">A35</font></span></a>, <a class="txt" href="#E1:66_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="ref" href="afinsq_2.html#D5" target="_self" title="AFINSQ_2:def.5">Def5</a></span>;<br/><a NAME="E12:66_1_4_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c8">k</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> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E13:66_1_4_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66_1_4_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A48</font></span></a>, <a class="txt" href="#E11:66_1_4_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E61">A50</font></span></a>, <a class="ref" href="afinsq_2.html#T44" target="_self" title="AFINSQ_2:th.44">Th44</a></span>;<br/><span class="kw">hence </span><a NAME="E14:66_1_4_1_2_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:66_1_4_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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="66_1_4_1_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:66_1_4_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A51</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</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"> contradiction</span><br/><div class="add"><a NAME="E2:66_1_4_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66_1_4_1"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E3:66_1_4_1_2_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:66_1_4_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E58">A52</font></span>: 
<font color="Maroon" title="c14">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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:66_1_4_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A53</font></span>: 
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</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> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <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="c15">x</font> = <font color="Maroon" title="c14">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1_4_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A52</font></span></a></span>;<br/><a NAME="E8:66_1_4_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c15">x</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1_4_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A52</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E9:66_1_4_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A54</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_4_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A51</font></span></a>, <a class="txt" href="#E6:66_1_4_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E59">A53</font></span></a>, <a class="ref" href="afinsq_2.html#T44" target="_self" title="AFINSQ_2:th.44">Th44</a></span>;<br/><a NAME="E10:66_1_4_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E61">A55</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E11:66_1_4_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_4_1"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E12:66_1_4_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A56</font></span>: 
<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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_4"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="txt" href="#E1:66_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="txt" href="#E10:66_1_4_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E61">A55</font></span></a>, <a class="ref" href="afinsq_2.html#D5" target="_self" title="AFINSQ_2:def.5">Def5</a></span>;<br/><a NAME="E13:66_1_4_1_2_1_1_1_1_2"/>
<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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c8">k</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E2:66_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E56">A45</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E14:66_1_4_1_2_1_1_1_1_2"/><span class="kw">then </span>
<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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66_1_4_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E59">A53</font></span></a>, <a class="txt" href="#E12:66_1_4_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E62">A56</font></span></a>, <a class="ref" href="afinsq_2.html#T44" target="_self" title="AFINSQ_2:th.44">Th44</a></span>;<br/><span class="kw">hence </span><a NAME="E15:66_1_4_1_2_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:66_1_4_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E60">A54</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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><span class="kw">hence </span><a NAME="E4:66_1_4_1_2_1_1"/>
contradiction
 ; <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="66_1_4_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:66_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E55">A57</font></span>: 
not <font color="Maroon" title="c8">k</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> <font color="Maroon" title="c2">Y</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/><div class="add"><a NAME="E2:66_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E56">A58</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E3:66_1_4_1_2_1_2"/>
 <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> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E55">A57</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:66_1_4_1_2_1_2"/><span class="kw">then </span>
 <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> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E56">A58</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:66_1_4_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_4_1"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</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>
<span class="kw">hence </span><a NAME="E12:66_1_4_1"/>
contradiction
 ; <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>
<a NAME="E11:66_1"/>
(  <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> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">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="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th31</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></span>;<br/>
<a NAME="E12:66_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A59</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="afinsq_2.html#T43" target="_self" title="AFINSQ_2:th.43">Th43</a></span>;<br/>
<a NAME="E13:66_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E12:66_1"><span class="lab"><font color="Green" title="E47">A59</font></span></a>, <a class="txt" href="#E10:66_1"><span class="lab"><font color="Green" title="E46">A31</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E14:66_1"/>
 <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <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> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E9:66_1"><span class="lab"><font color="Green" title="E45">A30</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</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">assume </span><a NAME="E2:66"/><span class="lab"><font color="Green" title="E40">A60</font></span>: 
 <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <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> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</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="c1">X</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c2">Y</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">m</font>, <font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="afinsq_2.html#D6" target="_self" title="AFINSQ_2:def.6">AFINSQ_2:def 6</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="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:66"/><span class="lab"><font color="Green" title="E41">A61</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E4:66"/><span class="lab"><font color="Green" title="E42">A62</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</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="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font></span><br/>

<a NAME="E5:66"/>
<font color="Maroon" title="c7">n</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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:66"><span class="lab"><font color="Green" title="E42">A62</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">then </span><span class="kw">consider </span><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:66"/><span class="lab"><font color="Green" title="E43">A63</font></span>: 
<font color="Maroon" title="c8">y</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> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:66"/><span class="lab"><font color="Green" title="E44">A64</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</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="c9">y</font> = <font color="Maroon" title="c8">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:66"><span class="lab"><font color="Green" title="E43">A63</font></span></a></span>;<br/>
<a NAME="E10:66"/><span class="lab"><font color="Green" title="E45">A65</font></span>: 
<font color="Maroon" title="c7">n</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c9">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E40">A60</font></span></a>, <a class="txt" href="#E7:66"><span class="lab"><font color="Green" title="E43">A63</font></span></a>, <a class="txt" href="#E8:66"><span class="lab"><font color="Green" title="E44">A64</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/>
<a NAME="E11:66"/>
<font color="Maroon" title="c9">y</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> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:66"><span class="lab"><font color="Green" title="E43">A63</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E12:66"/><span class="kw">then </span>
<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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E13:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A66</font></span>: 
<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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c9">y</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E40">A60</font></span></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></span>;<br/>
<a NAME="E14:66"/><span class="lab"><font color="Green" title="E47">A67</font></span>: 
 <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> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c9">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E15:66"/>
<font color="Maroon" title="c6">m</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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E41">A61</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">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:66"/><span class="lab"><font color="Green" title="E48">A68</font></span>: 
<font color="Maroon" title="c10">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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E18:66"/><span class="lab"><font color="Green" title="E49">A69</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</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="c11">x</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:66"><span class="lab"><font color="Green" title="E48">A68</font></span></a></span>;<br/>
<a NAME="E20:66"/>
<font color="Maroon" title="c11">x</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> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:66"><span class="lab"><font color="Green" title="E48">A68</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E21:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A70</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c9">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:66"><span class="lab"><font color="Green" title="E47">A67</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E22:66"/>
<font color="Maroon" title="c6">m</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="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E40">A60</font></span></a>, <a class="txt" href="#E17:66"><span class="lab"><font color="Green" title="E48">A68</font></span></a>, <a class="txt" href="#E18:66"><span class="lab"><font color="Green" title="E49">A69</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:66"/>
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66"><span class="lab"><font color="Green" title="E45">A65</font></span></a>, <a class="txt" href="#E21:66"><span class="lab"><font color="Green" title="E50">A70</font></span></a>, <a class="txt" href="#E13:66"><span class="lab"><font color="Green" title="E46">A66</font></span></a>, <a class="ref" href="afinsq_2.html#D5" target="_self" title="AFINSQ_2:def.5">Def5</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>
