<?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>let </b><font color="Maroon" title="c3">n1</font>, <font color="Maroon" title="c4">n2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:65"/><i><font color="Green" title="E50">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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c5">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="c6">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>for <font color="Olive" title="b1">n1</font>, <font color="Olive" title="b2">n2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> $1 holds <br/><font color="Olive" title="b1">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Olive" title="b2">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1;<br/>
<a NAME="E2:65"/><i><font color="Green" title="E51">A2</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">n1</font>, <font color="Maroon" title="c8">n2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:65_1"/><i><font color="Green" title="E52">A3</font></i>: 
( <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c8">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">u</font> = <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> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="real_3.html#T38" target="_self" title="REAL_3:th.38">Th38</a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E3:65_1"/>
<font color="Maroon" title="c7">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="txt" href="real_3.html#E1:65_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="real_3.html#D6" target="_self" title="REAL_3:def.6">Def6</a></i>;<br/>
<a NAME="E4:65_1"/><b>then </b><i><font color="Green" title="E53">A4</font></i>: 
<font color="Maroon" title="c7">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="real_3.html#E1:65_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="real_3.html#D6" target="_self" title="REAL_3:def.6">Def6</a></i>;<br/>
<a NAME="E5:65_1"/>
1 <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c8">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T64" title="NEWTON:th.64">NEWTON:64</a></i>;<br/>
<b>hence </b><a NAME="E6:65_1"/>
<font color="Maroon" title="c7">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c8">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="real_3.html#E4:65_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="euler_1.html#T9" title="EULER_1:th.9">EULER_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:65"/><i><font color="Green" title="E52">A5</font></i>: 
for <font color="Olive" title="b1">k</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">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:65_2"/><i><font color="Green" title="E53">A6</font></i>: 
for <font color="Olive" title="b1">n1</font>, <font color="Olive" title="b2">n2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> <font color="Maroon" title="c7">k</font> holds <br/><font color="Olive" title="b1">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Olive" title="b2">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">n1</font> = <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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="real_3.html#E1:65"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="real_3.html#T44" target="_self" title="REAL_3:th.44">Th44</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">n2</font> = <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> <font color="Maroon" title="c7">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="real_3.html#E1:65"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="real_3.html#T44" target="_self" title="REAL_3:th.44">Th44</a></i>;<br/>
<a NAME="E4:65_2"/><i><font color="Green" title="E54">A7</font></i>: 
<font color="Maroon" title="c8">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c9">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="real_3.html#E1:65_2"><i><font color="Green" title="E53">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">n3</font> = <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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="real_3.html#E1:65"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="real_3.html#T44" target="_self" title="REAL_3:th.44">Th44</a></i>;<br/>
<a NAME="E6:65_2"/>
<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">n4</font> = <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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="real_3.html#T38" target="_self" title="REAL_3:th.38">Th38</a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E8:65_2"/>
<font color="Maroon" title="c10">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n4</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n2</font>
 
<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="E9:65_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="real_3.html#E4:65_2"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="euler_1.html#T9" title="EULER_1:th.9">EULER_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:65"/>
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#E2:65"><i><font color="Green" title="E51">A2</font></i></a>, <a class="txt" href="real_3.html#E3:65"><i><font color="Green" title="E52">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:65"/>
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" title="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c3">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c4">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>


</div>
