<?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">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>set </b><font color="Maroon">r</font> = 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font>;<br/>
<b>assume </b><a NAME="E1:53"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>

<a NAME="E2:53"/><b>then </b><i><font color="Green">E32</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<a NAME="E3:53"/><i><font color="Green">E33</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="real_3.html#T5" target="_self">Th5</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<i><font color="Green">E38</font></i>: <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> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="int_1.html#K1">/]</a></span> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="txt" href="real_3.html#E2:53"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="real_3.html#T6" target="_self">Th6</a>, <a class="ref" href="real_3.html#T7" target="_self">Th7</a>, </i>
;<br/>
<i><font color="Green">E48</font></i>: <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0 = 
<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> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="txt" href="real_3.html#E3:53"><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="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="txt" href="real_3.html#E2:53"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="real_3.html#T6" target="_self">Th6</a>, <a class="ref" href="real_3.html#T7" target="_self">Th7</a>, </i>
;<br/>
<b>thus </b><i><font color="Green">E49</font></i>: <span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1 = 
<span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
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> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E2:53"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0</span>)</span>
<b>by </b><i>, <a class="txt" href="real_3.html#E2:53"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>
;<br/>

<a NAME="E7:53"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">i</font></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i>, <a class="txt" href="real_3.html#E3:53"><i><font color="Green">E33</font></i></a></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"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0 );<br/>
<a NAME="E8:53"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
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> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E2:53"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="int_1.html#T47">INT_1:47</a></i>
<br/>.= 
0

;<br/>

<b>hence </b><span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default">0</span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="txt" href="real_3.html#E3:53"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
0

;<br/><br/>


</div><b>end;</b></div>
<a NAME="E9:53"/><i><font color="Green">E58</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="53_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_5"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>

<br/>.= 
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> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></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></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E2:53"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K5">-</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E5:53"><i><font color="Green">E48</font></i></a>, <a class="ref" href="int_1.html#T47">INT_1:47</a></i>
<br/>.= 
0

;<br/>

<b>hence </b><span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default">0</span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="txt" href="real_3.html#E3:53"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
0

;<br/><br/>


</div><b>end;</b></div>
<a NAME="E10:53"/>
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#T8" target="_self">Th8</a>);<br/></i>
<b>hence </b><a NAME="E11:53"/>
( <span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></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> 0 &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></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> 0 )
 <b>by </b><i><a class="ref" href="real_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="int_1.html#T47">INT_1:47</a></i>;<br/>


</div>
