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

<b>let </b><font color="Maroon">p</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">Fr</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:75"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fr</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_1_1"/>
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 or  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1"><b>suppose </b></a><a NAME="E1:75_1_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:75_1_1_1"/><b>then </b>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E3:75_1_1_1"/><b>then </b>
( <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K47">addnat</a>  &amp; 0 <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K47">addnat</a>  )
 <b>by </b><i><a class="ref" href="stirl2_1.html#D3">STIRL2_1:def 3</a>, <a class="ref" href="binop_2.html#T5">BINOP_2:5</a></i>;<br/><b>hence </b><a NAME="E4:75_1_1_1"/>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font>
 <b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2"><b>suppose </b></a><a NAME="E1:75_1_1_2"/>
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">d1</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> implies  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> );<br/><a NAME="E3:75_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1_1_2_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:75_1_1_2_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 
;<br/>
<a NAME="E3:75_1_1_2_1"/><b>then </b>
( 0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</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="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E4:75_1_1_2_1"/><b>then </b>
1 <a href="tarski.html#R1">c=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E5:75_1_1_2_1"/><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#K4">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E6:75_1_1_2_1"/><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#K4">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="catalan2.html#K5" target="_self">.</a> 0 )
 
<b>by </b><i><a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E7:75_1_1_2_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K9">&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#K4">|</a> 1</span>)</span> <a href="stirl2_1.html#K8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> &amp;  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="stirl2_1.html#K8">.</a> 0 )
 
<b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, <a class="ref" href="afinsq_1.html#T38">AFINSQ_1:38</a></i>;<br/>
<b>hence </b><a NAME="E8:75_1_1_2_1"/>
 <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:75_1_1_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">n</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">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_2_2"><b>proof </b></a><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>assume </b><a NAME="E1:75_1_1_2_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>set </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>assume </b><a NAME="E2:75_1_1_2_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E3:75_1_1_2_2"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:75_1_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E5:75_1_1_2_2"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, , <a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="txt" href="catalan2.html#E1:75"><i><font color="Green">E51</font></i></a>, <a class="ref" href="card_fin.html#T44">CARD_FIN:44</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<b>hence </b><a NAME="E6:75_1_1_2_2"/>
 <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:75_1_1_2"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">n</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">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><a NAME="E6:75_1_1_2"/>
<font color="Maroon">d1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d1</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:75_1_1_2"/><b>then </b>
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> &amp; <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> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">d1</font> <a href="xcmplx_0.html#K2">+</a> 1 <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="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/><b>hence </b><a NAME="E8:75_1_1_2"/>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font>
 <b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:75"/>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font>
 ;<br/>


</div>
