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

<b>let </b><font color="Maroon" title="c2">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:107"/><i><font color="Green" title="E64">A1</font></i>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon" title="c3">s1</font> =  <a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font>;<br/>
<b>set </b><font color="Maroon" title="c4">s</font> =  <a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="real_3.html#NK14" title="REAL_3:NK.14">bcf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default">$1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default">$1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">s3</font> =  <a href="real_3.html#NK14" title="REAL_3:NK.14">bcf</a> <font color="Maroon" title="c2">r</font>;<br/>
<span class="p1">(<span class="default"><a href="real_3.html#NK14" title="REAL_3:NK.14">bcf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK14" title="REAL_3:NK.14">bcf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D9" target="_self" title="REAL_3:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D9" target="_self" title="REAL_3:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E1:107"><i><font color="Green" title="E64">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T114" title="XCMPLX_1:th.114">XCMPLX_1:114</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D6" target="_self" title="REAL_3:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D6" target="_self" title="REAL_3:def.6">Def6</a></i>
;<br/>
<a NAME="E3:107"/><b>then </b><i><font color="Green" title="E65">A2</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
;<br/>
<a NAME="E4:107"/><i><font color="Green" title="E66">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E1:107_2"/><i><font color="Green" title="E67">A4</font></i>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="real_3.html#T45" target="_self" title="REAL_3:th.45">Th45</a>, <a class="txt" href="real_3.html#E1:107"><i><font color="Green" title="E64">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:107_2"/><i><font color="Green" title="E68">A5</font></i>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK14" title="REAL_3:NK.14">bcf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
 ;<br/>

<span class="p1">(<span class="default"><a href="real_3.html#NK14" title="REAL_3:NK.14">bcf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK14" title="REAL_3:NK.14">bcf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D9" target="_self" title="REAL_3:def.9">Def9</a></i>
<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" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E2:107_2"><i><font color="Green" title="E68">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T57" title="XCMPLX_1:th.57">XCMPLX_1:57</a></i>
<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" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E1:107_2"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="xcmplx_1.html#T114" title="XCMPLX_1:th.114">XCMPLX_1:114</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D6" target="_self" title="REAL_3:def.6">Def6</a></i>
;<br/>
<b>hence </b><a NAME="E4:107_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:107"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="real_3.html#E3:107"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="real_3.html#E4:107"><i><font color="Green" title="E66">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:107"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="real_3.html#NK14" title="REAL_3:NK.14">bcf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK9" title="REAL_3:NK.9">c_n</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 ;<br/>


</div>
