<?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>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"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0;<br/>
<a NAME="E1:60"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<i><font color="Green">E32</font></i>: <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 = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="txt" href="real_3.html#E5:60_1"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="txt" href="real_3.html#E1:60_1"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <font color="Maroon">r</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="txt" href="real_3.html#E1:60_1"><i><font color="Green">E32</font></i></a></i>
;<br/>
<a NAME="E2:60_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">r</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E3:60_1"/><b>then </b>
 <a href="int_1.html#K4">frac</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E4:60_1"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="int_1.html#K4">frac</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T138">XREAL_1:138</a></i>;<br/>
<a NAME="E5:60_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="int_1.html#K4">frac</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E6:60_1"/>
<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 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="int_1.html#K4">frac</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E7:60_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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#K5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="real_3.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T16">XREAL_1:16</a></i>;<br/>
<a NAME="E8:60_1"/><b>then </b>
<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#K5">-</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E9:60_1"/><b>then </b>
<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 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E10:60_1"/>
<font color="Maroon">S<sub>2</sub></font>[1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:60"/><i><font color="Green">E38</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="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <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="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_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="E1:60_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<i><font color="Green">E48</font></i>: <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> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i/>
;<br/>
<a NAME="E3:60_2"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E4:60_2"/><b>then </b>
 <a href="int_1.html#K4">frac</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E5:60_2"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T138">XREAL_1:138</a></i>;<br/>
<a NAME="E6:60_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E7:60_2"/>
<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> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="real_3.html#T8" target="_self">Th8</a>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E8:60_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="real_3.html#E2:60_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xreal_1.html#T16">XREAL_1:16</a></i>;<br/>
<a NAME="E9:60_2"/><b>then </b>
<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> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E10:60_2"/><b>then </b>
<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> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E11:60_2"/>
(  not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:60"/><i><font color="Green">E50</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="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(, );<br/></i>
<a NAME="E4:60"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E5:60"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 implies <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="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="real_3.html#T8" target="_self">Th8</a></i>;<br/>


</div>
