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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p'</font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_1_1"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">n</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font> )
 <b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E3:4_1_1_1"/><b>then </b>
( <font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp;  <a href="xboole_0.html#K1">{}</a>  is   <a href="afinsq_1.html#NM1">XFinSequence</a> )
 <b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a>, <a class="ref" href="afinsq_1.html#T32">AFINSQ_1:32</a></i>;<br/><b>hence </b><a NAME="E4:4_1_1_1"/>
 ex <font color="Olive">q'</font> being  <a href="afinsq_1.html#NM1">XFinSequence</a> st <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">n</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Olive">q'</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">q'</font> being  <a href="afinsq_1.html#NM1">XFinSequence</a> st <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <font color="Olive">k</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Olive">q'</font>;<br/><a NAME="E2:4_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:4_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E2:4_1_1_2_1"/><b>then </b>
( <font color="Maroon">p'</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp;  <a href="xboole_0.html#K1">{}</a>  is   <a href="afinsq_1.html#NM1">XFinSequence</a> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a>, <a class="ref" href="afinsq_1.html#T32">AFINSQ_1:32</a></i>;<br/>
<b>hence </b><a NAME="E3:4_1_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div><a NAME="E3:4_1_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font> <a href="binop_2.html#K23">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:4_1_1_2_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>]
 ;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:4_1_1_2_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">q'</font> being   <a href="afinsq_1.html#NM1">XFinSequence</a><b> such that </b><br/><a NAME="E4:4_1_1_2_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:4_1_1_2_2"/>
( <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 &amp; <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T12">XREAL_1:12</a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E6:4_1_1_2_2"/><b>then </b>
( ( <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font> ) or <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font> )
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E7:4_1_1_2_2"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="afinsq_1.html#T14">AFINSQ_1:14</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:4_1_1_2_2"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> is   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E9:4_1_1_2_2"/><b>then </b>
( <font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span> <a href="ordinal4.html#K1">^</a> <span class="p1"><a href="afinsq_1.html#K6">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="afinsq_1.html#K6">%&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="card_fin.html#T43">CARD_FIN:43</a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E10:4_1_1_2_2"/><b>then </b>
( <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span> <a href="ordinal4.html#K1">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K6">&lt;%</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="afinsq_1.html#K6">%&gt;</a></span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span> &amp; <span class="p1"><a href="afinsq_1.html#K6">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="afinsq_1.html#K6">%&gt;</a></span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font> is   <a href="afinsq_1.html#NM1">XFinSequence</a> )
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#T30">AFINSQ_1:30</a></i>;<br/>
<b>hence </b><a NAME="E11:4_1_1_2_2"/>
 ex <font color="Olive">q'</font> being  <a href="afinsq_1.html#NM1">XFinSequence</a> st <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Olive">q'</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E4:4_1_1_2"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><b>reconsider </b><font color="Maroon">n1</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="nat_1.html#T61">NAT_1:61</a></i>;<br/><b>reconsider </b><font color="Maroon">n2</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E7:4_1_1_2"/>
 ex <font color="Olive">q'</font> being  <a href="afinsq_1.html#NM1">XFinSequence</a> st <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">n2</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Olive">q'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:4_1_1_2"/>
 ex <font color="Olive">q'</font> being  <a href="afinsq_1.html#NM1">XFinSequence</a> st <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">n</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Olive">q'</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:4"/>
 ex <font color="Olive">q'</font> being  <a href="afinsq_1.html#NM1">XFinSequence</a> st <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">n</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Olive">q'</font>
 ;<br/>


</div>
