<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c2">b</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) holds <br/>for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) implies for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">s</font> =  <a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">s2</font> =  <a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:73"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/>

<a NAME="E2:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 
;<br/>
<a NAME="E3:73"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/>
<a NAME="E4:73"/>
( <span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_3.html#T38" target="_self" title="REAL_3:th.38">Th38</a></span>;<br/>
<a NAME="E5:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/>
<span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_3.html#D7" target="_self" title="REAL_3:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_3.html#D7" target="_self" title="REAL_3:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_3.html#D7" target="_self" title="REAL_3:def.7">Def7</a></span>
;<br/>
<a NAME="E7:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A5</font></span>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/>

<a NAME="E8:73"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T41" title="XREAL_1:th.41">XREAL_1:41</a></span>;<br/>
<a NAME="E9:73"/><span class="kw">then </span>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T95" title="SQUARE_1:th.95">SQUARE_1:95</a></span>;<br/>
<a NAME="E10:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A6</font></span>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></span>;<br/>
<a NAME="E11:73"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E12:73"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<a NAME="E13:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A7</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/>
<a NAME="E14:73"/><span class="lab"><font color="Green" title="E58">A8</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">n</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:73_2"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:73_2"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2]</span><br/>

<span class="lab"><font color="Green" title="E61">A11</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>

;<br/>
<a NAME="E4:73_2"/>
<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E5:73_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A12</font></span>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_3.html#T38" target="_self" title="REAL_3:th.38">Th38</a></span>;<br/>
<span class="lab"><font color="Green" title="E63">A13</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T2" title="WSIERP_1:th.2">WSIERP_1:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>

;<br/>
<span class="lab"><font color="Green" title="E64">A14</font></span>: <span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_3.html#D7" target="_self" title="REAL_3:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>

;<br/>
<a NAME="E8:73_2"/>
( <span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="real_3.html#T51" target="_self" title="REAL_3:th.51">Th51</a></span>;<br/>
<a NAME="E9:73_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="txt" href="#E5:73_2"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:73_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_2"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E7:73_2"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="txt" href="#E3:73_2"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="txt" href="#E6:73_2"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:73"/>
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:73"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<a NAME="E16:73"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E17:73"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E18:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A15</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T2" title="WSIERP_1:th.2">WSIERP_1:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2

;<br/>
<a NAME="E20:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A16</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[1]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E18:73"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E21:73"/>
<span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_3.html#D7" target="_self" title="REAL_3:def.7">Def7</a></span>;<br/>
<a NAME="E22:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A17</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E13:73"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E23:73"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="fib_num.html#S1" title="FIB_NUM:sch.1">FIB_NUM:sch 1</a>(<a class="txt" href="#E22:73"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="txt" href="#E20:73"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E14:73"><span class="lab"><font color="Green" title="E58">A8</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E24:73"/>
<span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
