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

<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>for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">Catal</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">Catal</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="newton.html#K2">|^</a> <font color="Olive">n</font></span>)</span> ) &amp; (  <a href="complex1.html#K18">abs</a> <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 implies ( <font color="Olive">Catal</font> is <a href="series_1.html#V2">absolutely_summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Olive">Catal</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Olive">Catal</font></span>)</span> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <font color="Olive">Catal</font> ) ) );<br/>
<a NAME="E1:76"/><i><font color="Green">E51</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#K1">REAL</a>  holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </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="E1:76_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">Cat</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:76_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>take </b>
 <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font>
;<br/>

<b>thus </b><a NAME="E5:76_1"/>
 <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<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="E6:76_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<b>take </b>
<font color="Maroon">Cat</font>
;<br/>

<b>thus </b><a NAME="E7:76_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="newton.html#K2">|^</a> <font color="Olive">n</font></span>)</span>
 <b>by </b><i>, </i>;<br/>

<b>assume </b><a NAME="E8:76_1"/><i><font color="Green">E57</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4
 ;<br/>

<b>set </b><font color="Maroon">a</font> = 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">G</font> = <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="prepower.html#K2">GeoSeq</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">Cat</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E9:76_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:76_1_1"/>
<font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> 0</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:76_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">Cat</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> 0</span>)</span> &amp; <font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> 0 <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> 0 <a href="hidden.html#R1">=</a> 1 &amp;  <a href="complex1.html#K18">abs</a> 1 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a>, <a class="ref" href="catalan1.html#T11">CATALAN1:11</a>, <a class="ref" href="newton.html#T9">NEWTON:9</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:76_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i><a class="ref" href="prepower.html#D1">PREPOWER:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:76_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:76_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>set </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<a NAME="E2:76_1_2"/>
(  <a href="catalan1.html#K1">Catalan</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="catalan1.html#K1">Catalan</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="catalan1.html#T17">CATALAN1:17</a></i>;<br/>
<a NAME="E3:76_1_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="catalan1.html#K1">Catalan</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="catalan1.html#K1">Catalan</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E4:76_1_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan1.html#T21">CATALAN1:21</a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/>
<a NAME="E5:76_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/>
<a NAME="E6:76_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/>
<a NAME="E7:76_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:76_1_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</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="complex1.html#T132">COMPLEX1:132</a>, <a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/>
<a NAME="E9:76_1_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">Cat</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K24">*</a> 4 )
 
<b>by </b><i>, <a class="ref" href="seq_1.html#T16">SEQ_1:16</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E10:76_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">Cat</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">Cat</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">Cat</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T6" target="_self">Th6</a>, <a class="ref" href="seq_1.html#T16">SEQ_1:16</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E11:76_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">Cat</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E12:76_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1]
 <b>by </b><i><a class="ref" href="prepower.html#T4">PREPOWER:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:76_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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#S1">NAT_1:sch 1</a>(<a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="ref" href="catalan2.html#D1" target="_self">Def1</a>);<br/></i>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:76_1_3"/>
( <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">Cat</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E2:76_1_3"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">Cat</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/></div><b>end;</b></div>
<a NAME="E13:76_1"/>
( 4 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="complex1.html#K18">abs</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="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E14:76_1"/><b>then </b>
( 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 4</span>)</span> &amp; 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 <a href="binop_2.html#K24">*</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E15:76_1"/><b>then </b>
( 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E16:76_1"/><b>then </b>
( <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="prepower.html#K2">GeoSeq</a>  is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#T28">SERIES_1:28</a></i>;<br/>
<a NAME="E17:76_1"/><b>then </b>
 <a href="seq_1.html#K22">abs</a> <font color="Maroon">Cat</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="catalan2.html#T6" target="_self">Th6</a>, <a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>;<br/>
<b>hence </b><a NAME="E18:76_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">Cat</font> is <a href="series_1.html#V2">absolutely_summable</a>
 <b>by </b><i><a class="ref" href="series_1.html#D5">SERIES_1:def 5</a></i>;<br/>

<a NAME="E19:76_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="catalan2.html#K6" target="_self">(##)</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">Cat</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:76_1_4"/>
<font color="Maroon">y</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">y</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="binop_2.html#K23">+</a> 1;<br/>
<b>consider </b><font color="Maroon">Fr1</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:76_1_4"/><i><font color="Green">E95</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E5:76_1_4"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">Cat</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Olive">i</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:76_1_4"/><i><font color="Green">E97</font></i>: 
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="catalan2.html#K6" target="_self">(##)</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">Cat</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="catalan2.html#E4:76_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">Catal</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:76_1_4"/><i><font color="Green">E98</font></i>: 
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">Catal</font> <a href="hidden.html#R1">=</a>  <a href="catalan1.html#K1">Catalan</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E9:76_1_4"/><i><font color="Green">E99</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Catal</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E10:76_1_4"/><i><font color="Green">E129</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">Catal</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Olive">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">j</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E11:76_1_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Catal</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<a NAME="E12:76_1_4"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Catal</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">CatalR</font> = <font color="Maroon">Catal</font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">Catal</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">k</font></span>)</span>;<br/>
<a NAME="E14:76_1_4"/><i><font color="Green">E131</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:76_1_4_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">Catal</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
;<br/>

<b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font>, <font color="Maroon">n'</font> = <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>thus </b><a NAME="E3:76_1_4_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">Catal</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">Catal</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>] )
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">Fr2</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E16:76_1_4"/><i><font color="Green">E132</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E17:76_1_4"/><i><font color="Green">E133</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Maroon">Fr2</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6">STIRL2_1:sch 6</a>(<a class="ref" href="catalan2.html#T10" target="_self">Th10</a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:76_1_4_2"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr2</font>
 ;<br/><a NAME="E2:76_1_4_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="catalan2.html#T13" target="_self">Th13</a>, <a class="ref" href="catalan2.html#T11" target="_self">Th11</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:76_1_4_2"/>
<font color="Maroon">Fr2</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">CatalR</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="catalan2.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green">E135</font></i>:  <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">CatalR</font></span>)</span>
<b>by </b><i><a class="ref" href="catalan2.html#T9" target="_self">Th9</a>, <a class="ref" href="catalan2.html#T11" target="_self">Th11</a>, <a class="txt" href="catalan2.html#E6:76_1"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="catalan2.html#E14"><i><font color="Green">Lemma75</font></i></a>, <a class="txt" href="catalan2.html#E12:76_1"><i><font color="Green">E62</font></i></a></i>
<br/>.= 
<font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:76_1_4_4"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr2</font>
 ;<br/><a NAME="E2:76_1_4_4"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="catalan2.html#T11" target="_self">Th11</a>, , <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E3:76_1_4_4"/><b>then </b>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:76_1_4_4"/><b>then </b><i><font color="Green">E140</font></i>: 
( <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>then </b><font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">Cat</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="catalan2.html#T11" target="_self">Th11</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">Cat</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="catalan2.html#T15" target="_self">Th15</a>, <a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="catalan2.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="catalan2.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="catalan2.html#T15" target="_self">Th15</a>, <a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Catal</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="catalan2.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Catal</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Catal</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<font color="Maroon">Fr2</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="catalan2.html#T11" target="_self">Th11</a>, <a class="ref" href="catalan2.html#T12" target="_self">Th12</a>, </i>
;<br/><b>hence </b><a NAME="E6:76_1_4_4"/>
<font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fr2</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E21:76_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="catalan2.html#K6" target="_self">(##)</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">Cat</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="catalan2.html#E13"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="catalan2.html#T13" target="_self">Th13</a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="catalan2.html#T11" target="_self">Th11</a>, <a class="ref" href="afinsq_1.html#T10">AFINSQ_1:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:76_1"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">Cat</font> <a href="seqm_3.html#K1">^\</a> 1 <a href="funct_2.html#R2">=</a> <font color="Maroon">Cat</font> <a href="catalan2.html#K6" target="_self">(##)</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">Cat</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T8">SEQ_1:8</a></i>;<br/>
<a NAME="E21:76_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">Cat</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="series_1.html#T40">SERIES_1:40</a></i>;<br/>
<a NAME="E22:76_1"/><b>then </b>
( <font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">Cat</font> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">Cat</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font></span>)</span> &amp; <font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> 0</span>)</span> &amp; <font color="Maroon">r</font> <a href="prepower.html#K3">|^</a> 0 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="series_1.html#T13">SERIES_1:13</a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<a NAME="E23:76_1"/><b>then </b>
(  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Cat</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">Cat</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a>, <a class="txt" href="catalan2.html#E11:76_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="catalan1.html#T11">CATALAN1:11</a>, <a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<a NAME="E24:76_1"/><b>then </b>
 <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="series_1.html#T18">SERIES_1:18</a></i>;<br/>
<b>hence </b><a NAME="E25:76_1"/>
(  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font></span>)</span> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T2">WSIERP_1:2</a>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">SumC</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:76"/><i><font color="Green">E144</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#K1">REAL</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="ref" href="catalan2.html#T4" target="_self">Th4</a>);<br/></i>
<a NAME="E4:76"/><i><font color="Green">E148</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 &amp; not <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r</font></span>)</span> holds <br/><font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><b>proof </b></a><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>assume </b><a NAME="E1:76_2"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 )
 ;<br/>

<a NAME="E2:76_2"/>
<font color="Maroon">r</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">Catal</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:76_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">Catal</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E5:76_2"/><i><font color="Green">E150</font></i>: 
(  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 implies ( <font color="Maroon">Catal</font> is <a href="series_1.html#V2">absolutely_summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Catal</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Catal</font></span>)</span> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Catal</font> ) )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Catal</font>;<br/>
 <a href="series_1.html#K2">Sum</a> <font color="Maroon">Catal</font> = 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Catal</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Catal</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="ref" href="wsierp_1.html#T2">WSIERP_1:2</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Catal</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>
;<br/>
<a NAME="E7:76_2"/><b>then </b><i><font color="Green">E151</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Catal</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Catal</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E8:76_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="binop_2.html#K12">/</a> 4
 
<b>by </b><i>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<a NAME="E9:76_2"/><b>then </b>
4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 4</span>)</span> <a href="binop_2.html#K11">*</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E10:76_2"/><b>then </b><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E11:76_2"/>
(  <a href="quin_1.html#K1">delta</a> <font color="Maroon">r</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> &amp;  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="square_1.html#T59">SQUARE_1:59</a>, <a class="ref" href="square_1.html#T61">SQUARE_1:61</a></i>;<br/>
<b>hence </b><a NAME="E12:76_2"/>
( <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> or <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="ref" href="catalan2.html#D1" target="_self">Def1</a>, <a class="ref" href="catalan2.html#T6" target="_self">Th6</a>, <a class="ref" href="fib_num.html#T6">FIB_NUM:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:76"/><i><font color="Green">E153</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 holds <br/><font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_3"><b>proof </b></a><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>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="E1:76_3"/><i><font color="Green">E154</font></i>: 
( 0 <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#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 )
 ;<br/>

<a NAME="E2:76_3"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 )
 
<b>by </b><i><a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E3:76_3"/>
<font color="Maroon">s</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">Cs</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E5:76_3"/><i><font color="Green">E156</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">Cs</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="newton.html#K2">|^</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:76_3"/><i><font color="Green">E157</font></i>: 
(  <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 implies ( <font color="Maroon">Cs</font> is <a href="series_1.html#V2">absolutely_summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cs</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Cs</font></span>)</span> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cs</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E7:76_3"/>
<font color="Maroon">r</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">Cr</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E9:76_3"/><i><font color="Green">E158</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">Cr</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E10:76_3"/><i><font color="Green">E159</font></i>: 
(  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 implies ( <font color="Maroon">Cr</font> is <a href="series_1.html#V2">absolutely_summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cr</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Cr</font></span>)</span> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cr</font> ) )
 <b>by </b><i/>;<br/>
<div><i><font color="Green">E160</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:76_3_1"/>
( <font color="Maroon">s</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> &amp;  <a href="catalan1.html#K1">Catalan</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="ref" href="prepower.html#T17">PREPOWER:17</a>, <a class="ref" href="catalan1.html#T17">CATALAN1:17</a></i>;<br/><a NAME="E2:76_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Cr</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E3:76_3_1"/>
<font color="Maroon">Cs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Cr</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="catalan2.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:76_3"/>
( <font color="Maroon">Cs</font> is <a href="series_1.html#V1">summable</a> &amp; <font color="Maroon">Cr</font> is <a href="series_1.html#V1">summable</a> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="ref" href="catalan2.html#D1" target="_self">Def1</a>, <a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="series_1.html#T40">SERIES_1:40</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E13:76_3"/>
<font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="ref" href="catalan2.html#D1" target="_self">Def1</a>, <a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a>, <a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="tietze.html#T7">TIETZE:7</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:76"/><i><font color="Green">E161</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 holds <br/><span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">s</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_4"><b>proof </b></a><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>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="E1:76_4"/><i><font color="Green">E162</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <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> 1 <a href="binop_2.html#K12">/</a> 4 )
 ;<br/>

<a NAME="E2:76_4"/>
( 4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font> &amp; 4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 4</span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#D1" target="_self">Def1</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E3:76_4"/><b>then </b>
( 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a>, <a class="ref" href="xreal_1.html#T12">XREAL_1:12</a></i>;<br/>
<a NAME="E4:76_4"/><b>then </b>
(  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> &amp;  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="square_1.html#T93">SQUARE_1:93</a>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<a NAME="E5:76_4"/><b>then </b>
( 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> &amp; 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="binop_2.html#K24">*</a> 0 &amp; 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font> &amp; 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 0 )
 
<b>by </b><i><a class="ref" href="catalan2.html#D1" target="_self">Def1</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E6:76_4"/><b>then </b>
( <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> &amp; <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/>
<b>hence </b><a NAME="E7:76_4"/>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">R</font> = <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span> ) </span>}</span> ;<br/>
<a NAME="E7:76"/><i><font color="Green">E163</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_5"/>
<span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span> ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<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="E3:76_5"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:76_5"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> )
 <b>by </b><i><a class="ref" href="catalan2.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E6:76_5"/><i><font color="Green">E166</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_5_1"/>
not <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:76_5_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span> ) </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:76_5_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span> ) </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:76_5_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a></i>;<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="E7:76_5_1"/><i><font color="Green">E168</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; 0 <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> 1 <a href="binop_2.html#K12">/</a> 4 &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span> )
 ;<br/>
<a NAME="E8:76_5_1"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a>, <a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_5_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_5_1_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> or <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> )
 </span><b>by </b><i><a class="txt" href="catalan2.html#E13"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:76_5_1_1_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:76_5_1_1_1_1"/><b>then </b>
( <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a></i>;<br/><b>hence </b><a NAME="E3:76_5_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:76_5_1_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:76_5_1_1_1_2"/><b>then </b>
( <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> )
 <b>by </b><i>, <a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a></i>;<br/><b>hence </b><a NAME="E3:76_5_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:76_5_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">s</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E8:76_5"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">r</font> <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> 1 <a href="binop_2.html#K12">/</a> 4 )
 <b>by </b><i><a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/>
<a NAME="E9:76_5"/>
( <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span> ) </span>}</span>  &amp; <font color="Maroon">s</font> is   <a href="real_1.html#NM1">Real</a> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E10:76_5"/><b>then </b>
( <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E11:76_5"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a></i>;<br/>
<a NAME="E12:76_5"/>
( 4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 4</span>)</span> &amp; 4 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 4</span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E13:76_5"/><b>then </b>
( <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E14:76_5"/><b>then </b>
(  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="square_1.html#T93">SQUARE_1:93</a>, <a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a></i>;<br/>
<a NAME="E15:76_5"/><b>then </b>
( 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="binop_2.html#K10">-</a> 0 &amp; 1 <a href="binop_2.html#K23">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> &amp; 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="binop_2.html#K24">*</a> 0 &amp; 2 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="binop_2.html#K24">*</a> 0 &amp; 2 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a>, <a class="ref" href="xreal_1.html#T12">XREAL_1:12</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E16:76_5"/><b>then </b>
( <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> &amp; 1 <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a>, <a class="ref" href="xreal_1.html#T76">XREAL_1:76</a>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/>
<a NAME="E17:76_5"/><b>then </b>
( <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> &amp; 1 <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E18:76_5"/><b>then </b>
( <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E19:76_5"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:76"/><i><font color="Green">E172</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp;  <a href="complex1.html#K18">abs</a> <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 holds <br/><font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_6"><b>proof </b></a><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>assume </b><a NAME="E1:76_6"/><i><font color="Green">E173</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp;  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 )
 ;<br/>

<b>assume </b><a NAME="E2:76_6"/>
<font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E3:76_6"/><b>then </b>
( <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> &amp; <font color="Maroon">r</font> is   <a href="real_1.html#NM1">Real</a> &amp;  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E4:76_6"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <font color="Olive">r</font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="catalan2.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E5:76_6"/>
contradiction
 <b>by </b><i><a class="ref" href="catalan2.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:76"/><i><font color="Green">E174</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 holds <br/><font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_7"><b>proof </b></a><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>assume </b><a NAME="E1:76_7"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 )
 ;<br/>

<a NAME="E2:76_7"/>
<font color="Maroon">r</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">Cr</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:76_7"/><i><font color="Green">E177</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">Cr</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E5:76_7"/><i><font color="Green">E179</font></i>: 
(  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 implies ( <font color="Maroon">Cr</font> is <a href="series_1.html#V2">absolutely_summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cr</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Cr</font></span>)</span> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cr</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E6:76_7"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">CR</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E8:76_7"/><i><font color="Green">E180</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">CR</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E9:76_7"/><i><font color="Green">E181</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 implies ( <font color="Maroon">CR</font> is <a href="series_1.html#V2">absolutely_summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">CR</font></span>)</span> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">CR</font> ) )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_7_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="E1:76_7_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/><span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T12">NEWTON:12</a></i>
;<br/><b>then </b><i><font color="Green">E182</font></i>:  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
1 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_2.html#T1">SERIES_2:1</a></i>
;<br/><a NAME="E5:76_7_1"/>
0 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K10">-</a> 0
 <b>by </b><i><a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="xreal_1.html#T12">XREAL_1:12</a></i>;<br/><a NAME="E6:76_7_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="catalan1.html#K1">Catalan</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="power.html#T3">POWER:3</a>, <a class="ref" href="catalan1.html#T17">CATALAN1:17</a></i>;<br/><a NAME="E7:76_7_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="catalan1.html#K1">Catalan</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>then </b><font color="Maroon">CR</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="catalan2.html#E13"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="catalan2.html#T9" target="_self">Th9</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Cr</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">Cr</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
;<br/><b>hence </b><a NAME="E9:76_7_1"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">Cr</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CR</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E11:76_7"/><b>then </b><i><font color="Green">E183</font></i>: 
(  <a href="seq_1.html#K22">abs</a> <font color="Maroon">Cr</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">CR</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Cr</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">Cr</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="seq_1.html#T8">SEQ_1:8</a>, <a class="ref" href="tietze.html#T8">TIETZE:8</a></i>;<br/>
<b>assume </b><a NAME="E12:76_7"/>
<font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E13:76_7"/><b>then </b><i><font color="Green">E184</font></i>: 
( <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> &amp; 0 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K10">-</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="xreal_1.html#T12">XREAL_1:12</a>, , <a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/>
<a NAME="E14:76_7"/><b>then </b><i><font color="Green">E185</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">CR</font> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T6" target="_self">Th6</a>, <a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a>, <a class="txt" href="catalan2.html#E14"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="catalan2.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E15:76_7"/>
( 1 <a href="binop_2.html#K12">/</a> 4 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, , <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<a NAME="E16:76_7"/><b>then </b>
( 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 4</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span> &amp; 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 4</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E17:76_7"/><b>then </b>
( 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> &amp; 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E18:76_7"/><b>then </b>
(  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="square_1.html#T93">SQUARE_1:93</a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E19:76_7"/><b>then </b>
( 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="binop_2.html#K10">-</a> 0 &amp; 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="binop_2.html#K23">+</a> 0 &amp; 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K23">+</a> 0 &amp; 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="binop_2.html#K24">*</a> 0 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a>, <a class="ref" href="xreal_1.html#T12">XREAL_1:12</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E20:76_7"/><b>then </b>
( 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> &amp; 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> &amp; 0 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K10">-</a> 0 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="xreal_1.html#T12">XREAL_1:12</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E21:76_7"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a></i>;<br/>
<b>hence </b><a NAME="E22:76_7"/>
contradiction
 <b>by </b><i><a class="ref" href="catalan2.html#T10" target="_self">Th10</a>, <a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>;<br/>


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

<a NAME="E10:76"/>
<font color="Maroon">r</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">Cat</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E12:76"/><i><font color="Green">E187</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E13:76"/><i><font color="Green">E188</font></i>: 
(  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4 implies ( <font color="Maroon">Cat</font> is <a href="series_1.html#V2">absolutely_summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font></span>)</span> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon">SumC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font> ) )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">Cat</font>
;<br/>

<b>thus </b><a NAME="E14:76"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">Cat</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Olive">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a></i>;<br/>

<b>assume </b><a NAME="E15:76"/><i><font color="Green">E189</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> 4
 ;<br/>

<b>hence </b><a NAME="E16:76"/>
( <font color="Maroon">Cat</font> is <a href="series_1.html#V2">absolutely_summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font></span>)</span> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a></i>;<br/>

<a NAME="E17:76"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_8"/>
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:76_8"/><b>then </b>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 
;<br/>
<b>hence </b><a NAME="E3:76_8"/>
 <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="catalan2.html#T6" target="_self">Th6</a>, <a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a>, <a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">s</font> =  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_9_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r</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="76_9_1_1"><b>suppose </b></a><a NAME="E1:76_9_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:76_9_1_1"/>
2 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font>
 <b>by </b><i><a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a>, <a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_9_1_2"><b>suppose </b></a><a NAME="E1:76_9_1_2"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:76_9_1_2"/><b>then </b><i><font color="Green">E192</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E3:76_9_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="binop_2.html#K12">/</a> 4
 <b>by </b><i><a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/><a NAME="E4:76_9_1_2"/><b>then </b>
4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E5:76_9_1_2"/><b>then </b><i><font color="Green">E193</font></i>: 
1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:76_9_1_2"/><b>then </b>
 <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/><a NAME="E7:76_9_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K23">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E8:76_9_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><b>then </b><span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T77">XCMPLX_1:77</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#T67">SQUARE_1:67</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a>, <a class="ref" href="square_1.html#T59">SQUARE_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T77">XCMPLX_1:77</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="catalan2.html#T9" target="_self">Th9</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/><b>hence </b><a NAME="E10:76_9_1_2"/>
 <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E13"><i><font color="Green">Lemma74</font></i></a>, <a class="txt" href="catalan2.html#E14"><i><font color="Green">Lemma75</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E19:76"/>
(  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; ( <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies  <a href="series_1.html#K2">Sum</a> <font color="Maroon">Cat</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="catalan2.html#E13"><i><font color="Green">Lemma74</font></i></a></i>;<br/>


</div>
