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

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>set </b><font color="Maroon">s</font> =  <a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">s1</font> =  <a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">s2</font> =  <a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font>;<br/>
<a NAME="E1:89"/><i><font color="Green">E31</font></i>: 
( <span class="p1">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:89"/><i><font color="Green">E32</font></i>: 
( <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 1 &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:89"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i>, , <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<a NAME="E4:89"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:89_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">l</font>
 ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#T13" target="_self">Th13</a>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/>
<b>hence </b><a NAME="E3:89_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:89"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="ref" href="real_3.html#D3" target="_self">Def3</a>, <a class="ref" href="real_3.html#T12" target="_self">Th12</a>);<br/></i>
<b>hence </b><a NAME="E6:89"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">n</font>
 ;<br/>


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