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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> ) )</span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> $1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">$1 <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:22"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> )</span><br/>

<a NAME="E2:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="irrat_1.html#NK1" title="IRRAT_1:NK.1">^</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="power.html#T39" title="POWER:th.39">POWER:39</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E21">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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>1</sub></font>[<font color="Maroon" title="c3">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ; <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>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><span class="kw">thus </span><a NAME="E2:22_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_1_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_1_1_1_1"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 ; <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>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:22_1_1_1_1"/>
<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><a NAME="E3:22_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="irrat_1.html#T12" target="_self" title="IRRAT_1:th.12">Th12</a></span>;<br/><a NAME="E4:22_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="irrat_1.html#K1" title="IRRAT_1:func.1">aseq</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <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="#E1:22_1_1_1_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="irrat_1.html#T14" target="_self" title="IRRAT_1:th.14">Th14</a></span>;<br/><a NAME="E5:22_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K1" title="IRRAT_1:func.1">aseq</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K1" title="IRRAT_1:func.1">aseq</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1_1_1_1"><span class="lab"><font color="Green" title="E24">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="E6:22_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="irrat_1.html#K1" title="IRRAT_1:func.1">aseq</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_1_1_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="irrat_1.html#T14" target="_self" title="IRRAT_1:th.14">Th14</a></span>;<br/><a NAME="E7:22_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K1" title="IRRAT_1:func.1">aseq</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T155" title="XREAL_1:th.155">XREAL_1:155</a></span>;<br/><a NAME="E8:22_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K1" title="IRRAT_1:func.1">aseq</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="irrat_1.html#T7" target="_self" title="IRRAT_1:th.7">Th7</a></span>;<br/><span class="kw">hence </span><a NAME="E9:22_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1_1_1_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E7:22_1_1_1_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_1_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">n</font>
 ; <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>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:22_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="irrat_1.html#NK1" title="IRRAT_1:NK.1">^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="irrat_1.html#D2" target="_self" title="IRRAT_1:def.2">Def2</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="irrat_1.html#NK1" title="IRRAT_1:NK.1">^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1_1_1_2"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="newton.html#D3" title="NEWTON:def.3">NEWTON:def 3</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="kw">hence </span><a NAME="E4:22_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E4:22"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="irrat_1.html#NK1" title="IRRAT_1:NK.1">^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="irrat_1.html#D2" target="_self" title="IRRAT_1:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:22"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E20">A2</font></span></a></span>; <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="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> )</span><br/>

<a NAME="E6:22"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="irrat_1.html#T11" target="_self" title="IRRAT_1:th.11">Th11</a>, <a class="ref" href="newton.html#T18" title="NEWTON:th.18">NEWTON:18</a></span>;<br/>
<a NAME="E7:22"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E6:22"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E21">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E8:22"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
<span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span>
 ; <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="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> )</span><br/>

<span class="kw">hence </span><a NAME="E9:22"/>
<span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="irrat_1.html#D5" target="_self" title="IRRAT_1:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> )</span><br/>

<span class="kw">hence </span><a NAME="E10:22"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="irrat_1.html#T3" target="_self" title="IRRAT_1:th.3">Th3</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/>


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