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

<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>thus </b><a NAME="E1:64"/>
( <font color="Maroon">r</font> is <a href="rat_1.html#V1">rational</a> implies  ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> holds <br/><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="Olive">m</font> <a href="hidden.html#R1">=</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">r</font> is <a href="rat_1.html#V1">rational</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<a NAME="E3:64_1"/>
 <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="int_1.html#T69">INT_1:69</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:64_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E6:64_1"/><i><font color="Green">E32</font></i>: 
 <a href="int_1.html#K4">frac</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>
<a NAME="E7:64_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E8:64_1"/>
 <a href="int_1.html#K4">frac</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="int_1.html#T69">INT_1:69</a></i>;<br/>
<a NAME="E9:64_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="real_3.html#T9" target="_self">Th9</a>, <a class="ref" href="real_2.html#T143">REAL_2:143</a></i>;<br/>
<b>set </b><font color="Maroon">fd</font> =  <a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">n</font>,<font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">fm</font> =  <a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">n</font>,<font color="Maroon">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">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&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">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:64_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">r</font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i>, <a class="ref" href="int_1.html#T71">INT_1:71</a></i>;<br/><b>take </b>
1
;<br/><b>let </b><font color="Maroon">a</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E3:64_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><a NAME="E4:64_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</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" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="real_3.html#T10" target="_self">Th10</a>, <a class="ref" href="real_3.html#T4" target="_self">Th4</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">E49</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:64_1_1_2"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</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" target="_self">modSeq</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="real_3.html#E5:64_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
;<br/><b>let </b><font color="Maroon">a</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E5:64_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E6:64_1_1_2"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:64_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E8:64_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E9:64_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E11:64_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div><i><font color="Green">E68</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">E69</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:64_1_1_2_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, <a class="ref" href="nat_2.html#T14">NAT_2:14</a></i>;<br/><b>hence </b><a NAME="E3:64_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="real_3.html#T11" target="_self">Th11</a>, <a class="txt" href="real_3.html#E11:64_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="real_3.html#T10" target="_self">Th10</a>, </i>;<br/></div><b>end;</b></div><a NAME="E13:64_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="real_3.html#D2" target="_self">Def2</a>, <a class="txt" href="real_3.html#E5:64_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><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</font> = 
<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="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>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font>
<b>by </b><i><a class="txt" href="real_3.html#E5:64_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="real_3.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font>
<b>by </b><i><a class="txt" href="real_3.html#E3:64_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xcmplx_1.html#T57">XCMPLX_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="real_3.html#T8" target="_self">Th8</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="real_3.html#T11" target="_self">Th11</a>, <a class="txt" href="real_3.html#E7:64_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="real_3.html#E11:64_1_1_2"><i><font color="Green">E59</font></i></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" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>given </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E2:64"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> holds <br/><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="Olive">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Olive">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 0 ) holds <br/><font color="Olive">s</font> is <a href="rat_1.html#V1">rational</a>;<br/>
<a NAME="E3:64"/><i><font color="Green">E109</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">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:64_2"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 holds <br/><span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:64_2"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="real_3.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E3:64_2"/>
<font color="Maroon">s</font> is <a href="rat_1.html#V1">rational</a>
 <b>by </b><i><a class="ref" href="real_3.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:64"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</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">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:64_3"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">m</font>]
 ;<br/>

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E2:64_3"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>set </b><font color="Maroon">B</font> = 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <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">s</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>;<br/>
<i><font color="Green">E114</font></i>: <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">s</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</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" target="_self">scf</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <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">s</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T52">XCMPLX_1:52</a></i>
<br/>.= 
<font color="Maroon">s</font>

;<br/>
<a NAME="E4:64_3"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> holds <br/><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> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</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">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:64_3_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:64_3_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>thus </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> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">s</font></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> 1</span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#T7" target="_self">Th7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="real_3.html#T10" target="_self">Th10</a>, </i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <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">s</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> as  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><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" target="_self">scf</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">B</font></span>)</span> is <a href="rat_1.html#V1">rational</a>
 
;<br/>
<b>hence </b><a NAME="E7:64_3"/>
<font color="Maroon">s</font> is <a href="rat_1.html#V1">rational</a>
 <b>by </b><i><a class="ref" href="real_3.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:64"/>
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#T9" target="_self">Th9</a>);<br/></i>
<b>hence </b><a NAME="E6:64"/>
<font color="Maroon">r</font> is <a href="rat_1.html#V1">rational</a>
 <b>by </b><i/>;<br/>


</div>
