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

<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">r3</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:59"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">r3</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">r3</font> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r3</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> );<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>set </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1;<br/><b>set </b><font color="Maroon">oi</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/><b>reconsider </b><font color="Maroon">u</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">u</font> = <font color="Maroon">u</font>;<br/><a NAME="E4:59_1"/>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E5:59_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">r3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E6:59_1"/><b>then </b>
<font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E7:59_1"/><b>then </b>
<font color="Maroon">r3</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E8:59_1"/><b>then </b>
<font color="Maroon">r3</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><a NAME="E9:59_1"/><b>then </b>
not <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E6:59"><i><font color="Green">E36</font></i></a>, <a class="txt" href="pscomp_1.html#E4:59"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E10:59_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><b>hence </b><a NAME="E11:59_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><b>thus </b><a NAME="E12:59_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">u</font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">seq</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:59"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E5:59"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E6:59"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">seq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">seq</font> = <font color="Maroon">seq</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">seq</font>
;<br/>

<b>thus </b><a NAME="E8:59"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:59_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:59_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">seq</font> &amp; <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:59_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">r3</font> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r3</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:59_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="pscomp_1.html#E5:59_2"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> 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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1;<br/>
<b>set </b><font color="Maroon">oi</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<a NAME="E7:59_2"/>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E8:59_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">r3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E9:59_2"/><b>then </b>
<font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E10:59_2"/><b>then </b>
<font color="Maroon">r3</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E8:59_2"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E11:59_2"/><b>then </b>
<font color="Maroon">r3</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
;<br/>
<a NAME="E12:59_2"/><b>then </b>
not <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E6:59"><i><font color="Green">E36</font></i></a>, <a class="txt" href="pscomp_1.html#E4:59"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:59_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:59_2"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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

<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</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="E1:59_3"/><i><font color="Green">E45</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 ;<br/><b>set </b><font color="Maroon">cp</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">p</font></span>)</span></span><a href="int_1.html#K2">\]</a></span>;<br/><a NAME="E2:59_3"/><i><font color="Green">E46</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E3:59_3"/><i><font color="Green">E49</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">p</font></span>)</span></span><a href="int_1.html#K2">\]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><a NAME="E4:59_3"/><i><font color="Green">E50</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">p</font></span>)</span></span><a href="int_1.html#K2">\]</a></span>
 <b>by </b><i>, <a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">cp</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">p</font></span>)</span></span><a href="int_1.html#K2">\]</a></span> 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#T16">INT_1:16</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">cp</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E6:59_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><b>consider </b><font color="Maroon">m'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:59_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">r3</font> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><font color="Maroon">m'</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r3</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><font color="Maroon">m'</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1;<br/><b>set </b><font color="Maroon">oi</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/><a NAME="E9:59_3"/>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E10:59_3"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">r3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E11:59_3"/><b>then </b>
<font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E12:59_3"/><b>then </b>
<font color="Maroon">r3</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q3</font> where <font color="Olive">q3</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q3</font> &amp; <font color="Olive">q3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E13:59_3"/><b>then </b>
<font color="Maroon">r3</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><a NAME="E14:59_3"/><b>then </b>
not <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E6:59"><i><font color="Green">E36</font></i></a>, <a class="txt" href="pscomp_1.html#E4:59"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E15:59_3"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E16:59_3"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q3</font> where <font color="Olive">q3</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q3</font> &amp; <font color="Olive">q3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E18:59_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E19:59_3"/>
<font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E20:59_3"/><b>then </b>
(  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r3</font> &amp; <font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:59_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E21:59_3"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r3</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/><a NAME="E22:59_3"/>
<font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:59_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E23:59_3"/><b>then </b><i><font color="Green">E56</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T152">REAL_2:152</a></i>;<br/><a NAME="E24:59_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E25:59_3"/><b>then </b><i><font color="Green">E57</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:59_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="real_2.html#T151">REAL_2:151</a></i>;<br/><a NAME="E26:59_3"/>
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> <font color="Maroon">cp</font>
 <b>by </b><i>, <a class="txt" href="pscomp_1.html#E9:59"><i><font color="Green">E44</font></i></a>, <a class="ref" href="real_2.html#T152">REAL_2:152</a></i>;<br/><a NAME="E27:59_3"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>;<br/><a NAME="E28:59_3"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E8:59_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E29:59_3"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E6:59_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E30:59_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r3</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:59_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="pscomp_1.html#E4:59_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:59"/>
<font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>

<b>hence </b><a NAME="E11:59"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font>
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


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