<?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>thus </b><a NAME="E1:64"/>
( <font color="Maroon" title="c2">r</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a> implies  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">n</font> holds <br/><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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon" title="c2">r</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">r</font> = <font color="Maroon" title="c2">r</font> as  <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<a NAME="E3:64_1"/>
 <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="int_1.html#T69" title="INT_1:th.69">INT_1:69</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:64_1"/><i><font color="Green" title="E49">A1</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E6:64_1"/><i><font color="Green" title="E50">A2</font></i>: 
 <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="ref" href="real_3.html#T7" target="_self" title="REAL_3:th.7">Th7</a></i>;<br/>
<a NAME="E7:64_1"/><i><font color="Green" title="E51">A3</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="real_3.html#E5:64_1"><i><font color="Green" title="E49">A1</font></i></a></i>;<br/>
<a NAME="E8:64_1"/>
 <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="int_1.html#T69" title="INT_1:th.69">INT_1:69</a></i>;<br/>
<a NAME="E9:64_1"/><b>then </b><i><font color="Green" title="E52">A4</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font>
 
<b>by </b><i><a class="txt" href="real_3.html#E6:64_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="real_3.html#E7:64_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="real_2.html#T143" title="REAL_2:th.143">REAL_2:143</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">fd</font> =  <a href="real_3.html#K2" title="REAL_3:func.2">divSeq</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c4">m</font>;<br/>
<b>set </b><font color="Maroon" title="c7">fm</font> =  <a href="real_3.html#K1" title="REAL_3:func.1">modSeq</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c4">m</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1"/>
( <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_1"/>
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:64_1_1_1"/><b>then </b><i><font color="Green" title="E53">A5</font></i>: 
<font color="Maroon" title="c3">r</font> is <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>
 <b>by </b><i><a class="txt" href="real_3.html#E6:64_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="int_1.html#T71" title="INT_1:th.71">INT_1:71</a></i>;<br/><b>take </b>
1
;<br/><b>let </b><font color="Maroon" title="c8">a</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E3:64_1_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/><a NAME="E4:64_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>hence </b><a NAME="E5:64_1_1_1"/>
<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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="real_3.html#E2:64_1_1_1"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="real_3.html#T30" target="_self" title="REAL_3:th.30">Th30</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_2"/><i><font color="Green" title="E53">A6</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c8">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:64_1_1_2"/><i><font color="Green" title="E54">A7</font></i>: 
<span class="p1">(<span class="default"><a href="real_3.html#K2" title="REAL_3:func.2">divSeq</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>and </b><br/><a NAME="E4:64_1_1_2"/>
<span class="p1">(<span class="default"><a href="real_3.html#K1" title="REAL_3:func.1">modSeq</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="real_3.html#T25" target="_self" title="REAL_3:th.25">Th25</a></i>;<br/><b>take </b>
<font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
;<br/><b>let </b><font color="Maroon" title="c9">a</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E5:64_1_1_2"/><i><font color="Green" title="E55">A8</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/><a NAME="E6:64_1_1_2"/>
<font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E7:64_1_1_2"/><b>then </b><i><font color="Green" title="E56">A9</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="real_3.html#E5:64_1_1_2"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:64_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E9:64_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="real_3.html#E5:64_1_1_2"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">a1</font> = <font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E11:64_1_1_2"/><i><font color="Green" title="E57">A10</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div><i><font color="Green" title="E58">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1_1_2_1"/><i><font color="Green" title="E59">A12</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:64_1_1_2_1"/>
( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 or <font color="Maroon" title="c5">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="real_3.html#E9:64_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="nat_2.html#T14" title="NAT_2:th.14">NAT_2:14</a></i>;<br/><b>hence </b><a NAME="E3:64_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="real_3.html#E3:64_1_1_2"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="real_3.html#E1:64_1_1_2_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="txt" href="real_3.html#E1:64_1_1_2"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="real_3.html#D3" target="_self" title="REAL_3:def.3">Def3</a></i>;<br/></div><b>end;</b></div><a NAME="E13:64_1_1_2"/><i><font color="Green" title="E59">A13</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">a1</font>
 <b>by </b><i><a class="txt" href="real_3.html#E7:64_1_1_2"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="real_3.html#E11:64_1_1_2"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">a</font> = 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a1</font>
<b>by </b><i><a class="txt" href="real_3.html#E11:64_1_1_2"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="real_3.html#T37" target="_self" title="REAL_3:th.37">Th37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a1</font>
<b>by </b><i><a class="txt" href="real_3.html#E6:64_1"><i><font color="Green" title="E50">A2</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"><a href="real_3.html#K2" title="REAL_3:func.2">divSeq</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a1</font>
<b>by </b><i><a class="ref" href="real_3.html#T41" target="_self" title="REAL_3:th.41">Th41</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="real_3.html#E3:64_1_1_2"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="real_3.html#E12:64_1_1_2"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="real_3.html#E13:64_1_1_2"><i><font color="Green" title="E59">A13</font></i></a>, <a class="ref" href="real_3.html#T17" target="_self" title="REAL_3:th.17">Th17</a></i>
;<br/><b>hence </b><a NAME="E15:64_1_1_2"/>
<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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>given </b><font color="Maroon" title="c3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E2:64"/><i><font color="Green" title="E49">A14</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">n</font> holds <br/><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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<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">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st ( for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> $1 holds <br/><span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) holds <br/><font color="Olive" title="b1">s</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>;<br/>
<a NAME="E3:64"/><i><font color="Green" title="E50">A15</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">s</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="E1:64_2"/><i><font color="Green" title="E51">A16</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 holds <br/><span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:64_2"/>
for <font color="Olive" title="b1">m</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#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="real_3.html#E1:64_2"><i><font color="Green" title="E51">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:64_2"/>
<font color="Maroon" title="c4">s</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>
 <b>by </b><i><a class="ref" href="real_3.html#T34" target="_self" title="REAL_3:th.34">Th34</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:64"/><i><font color="Green" title="E51">A17</font></i>: 
for <font color="Olive" title="b1">m</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">m</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:64_3"/><i><font color="Green" title="E52">A18</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">m</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c5">s</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="E2:64_3"/><i><font color="Green" title="E53">A19</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="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<b>set </b><font color="Maroon" title="c6">B</font> = 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0: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="c5">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span>;<br/>
<i><font color="Green" title="E54">A20</font></i>: <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="c5">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> = 
<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="c5">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0: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="c5">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T52" title="XCMPLX_1:th.52">XCMPLX_1:52</a></i>
<br/>.= 
<font color="Maroon" title="c5">s</font>

;<br/>
<a NAME="E4:64_3"/>
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="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">m</font> holds <br/><span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:64_3_2"/>
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">m</font>
 ;<br/>

<a NAME="E2:64_3_2"/><b>then </b><i><font color="Green" title="E55">A21</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font> = 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c5">s</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="c7">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E47"><i><font color="Green" title="E41">Lm8</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="real_3.html#E2:64_3"><i><font color="Green" title="E53">A19</font></i></a>, <a class="txt" href="real_3.html#E2:64_3_2"><i><font color="Green" title="E55">A21</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">B</font> = 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0: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="c5">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span> as  <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="real_3.html#E1:64_3"><i><font color="Green" title="E52">A18</font></i></a></i>;<br/>
<a NAME="E6:64_3"/>
<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="c5">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">B</font></span>)</span> is <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>
 
;<br/>
<b>hence </b><a NAME="E7:64_3"/>
<font color="Maroon" title="c5">s</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>
 <b>by </b><i><a class="txt" href="real_3.html#E3:64_3"><i><font color="Green" title="E54">A20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:64"/>
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#E3:64"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="real_3.html#E4:64"><i><font color="Green" title="E51">A17</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:64"/>
<font color="Maroon" title="c2">r</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>
 <b>by </b><i><a class="txt" href="real_3.html#E2:64"><i><font color="Green" title="E49">A14</font></i></a></i>;<br/>


</div>
