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

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">r</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> ;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E56">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ;<br/>

<b>assume </b><a NAME="E2:57"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<b>assume </b><a NAME="E3:57"/><i><font color="Green" title="E58">A3</font></i>: 
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 ;<br/>

<b>assume </b><a NAME="E4:57"/><i><font color="Green" title="E59">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<b>assume </b><a NAME="E5:57"/><i><font color="Green" title="E60">A5</font></i>: 
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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 ;<br/>

<a NAME="E6:57"/>
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 2
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:57"><i><font color="Green" title="E56">A1</font></i></a></i>;<br/>
<a NAME="E7:57"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E4:57"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E8:57"/><b>then </b>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T26" title="FINSEQ_1:th.26">FINSEQ_1:26</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:57"/><i><font color="Green" title="E61">A6</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c5">k</font> as    <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>  <b>by </b><i><a class="txt" href="nat_4.html#E10:57"><i><font color="Green" title="E61">A6</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">f0</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span>;<br/>
<a NAME="E12:57"/><i><font color="Green" title="E62">A7</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E13:57"/><i><font color="Green" title="E63">A8</font></i>: 
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">r</font>
 ;<br/>

<a NAME="E2:57_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E12:57"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></i>;<br/>
<a NAME="E3:57_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/>
<a NAME="E4:57_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/>
<a NAME="E5:57_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:57"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:57_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_4.html#T3" target="_self" title="NAT_4:th.3">Th3</a>, <a class="txt" href="nat_4.html#E12:57"><i><font color="Green" title="E62">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E64">A9</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">r</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="txt" href="nat_4.html#E13:57"><i><font color="Green" title="E63">A8</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="txt" href="nat_4.html#E4:57"><i><font color="Green" title="E59">A4</font></i></a></i>
;<br/>
<a NAME="E15:57"/><i><font color="Green" title="E65">A10</font></i>: 
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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">k</font> <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">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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> ;<br/>

<b>assume </b><a NAME="E1:57_3"/><i><font color="Green" title="E66">A11</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<a NAME="E2:57_3"/><b>then </b>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E4:57"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:57_3"/><b>then </b><i><font color="Green" title="E67">A12</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E14:57"><i><font color="Green" title="E64">A9</font></i></a>, <a class="txt" href="nat_4.html#E4:57"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:57_3"/><i><font color="Green" title="E68">A13</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:57"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="nat_4.html#E1:57_3"><i><font color="Green" title="E66">A11</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_3_1"/>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> or  ex <font color="Olive" title="b1">n0</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>  st <br/>( <font color="Olive" title="b1">n0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span> &amp; <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">n0</font> ) )
 </span><b>by </b><i><a class="txt" href="nat_4.html#E3:57_3"><i><font color="Green" title="E67">A12</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3_1_1"><b>suppose </b></a><a NAME="E1:57_3_1_1"/><i><font color="Green" title="E69">A14</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:57_3_1_1"/><b>then </b><i><font color="Green" title="E70">A15</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><i><font color="Green" title="E71">A16</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c8">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c8">k</font>
<b>by </b><i><a class="txt" href="nat_4.html#E1:57_3_1_1"><i><font color="Green" title="E69">A14</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="txt" href="nat_4.html#E2:57_3_1_1"><i><font color="Green" title="E70">A15</font></i></a></i>
;<br/><a NAME="E4:57_3_1_1"/><i><font color="Green" title="E72">A17</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>;<br/><a NAME="E5:57_3_1_1"/><i><font color="Green" title="E73">A18</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><a NAME="E6:57_3_1_1"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><a NAME="E7:57_3_1_1"/><b>then </b>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><a NAME="E8:57_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E9:57_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E5:57_3_1_1"><i><font color="Green" title="E73">A18</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E10:57_3_1_1"/><b>then </b>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>;<br/><a NAME="E11:57_3_1_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E4:57_3"><i><font color="Green" title="E68">A13</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E12:57_3_1_1"/><b>then </b><i><font color="Green" title="E74">A19</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c8">k</font> is    <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> 
 <b>by </b><i><a class="txt" href="nat_4.html#E4:57_3"><i><font color="Green" title="E68">A13</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E13:57_3_1_1"/><i><font color="Green" title="E75">A20</font></i>: 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><a NAME="E14:57_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><a NAME="E15:57_3_1_1"/><b>then </b>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><a NAME="E16:57_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E17:57_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E13:57_3_1_1"><i><font color="Green" title="E75">A20</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E18:57_3_1_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E4:57_3_1_1"><i><font color="Green" title="E72">A17</font></i></a>, <a class="txt" href="nat_4.html#E4:57_3"><i><font color="Green" title="E68">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E19:57_3_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <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">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:57_3_1_1"><i><font color="Green" title="E71">A16</font></i></a>, <a class="txt" href="nat_4.html#E12:57_3_1_1"><i><font color="Green" title="E74">A19</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3_1_2"><b>suppose </b></a><a NAME="E1:57_3_1_2"/>
 ex <font color="Olive" title="b1">n0</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>  st <br/>( <font color="Olive" title="b1">n0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span> &amp; <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">n0</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">n0</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> <b> such that </b><br/><a NAME="E3:57_3_1_2"/><i><font color="Green" title="E69">A21</font></i>: 
( <font color="Maroon" title="c9">n0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span> &amp; <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c9">n0</font> )
 ;<br/><a NAME="E4:57_3_1_2"/><i><font color="Green" title="E70">A22</font></i>: 
<font color="Maroon" title="c9">n0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:57_3_1_2"><i><font color="Green" title="E69">A21</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E5:57_3_1_2"/><i><font color="Green" title="E71">A23</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c9">n0</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:57_3_1_2"><i><font color="Green" title="E69">A21</font></i></a>, <a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">k1</font> = <font color="Maroon" title="c8">k</font> as    <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>  ;<br/><a NAME="E7:57_3_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n0</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="nat_4.html#E4:57_3_1_2"><i><font color="Green" title="E70">A22</font></i></a></i>;<br/><a NAME="E8:57_3_1_2"/><b>then </b>
1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n0</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E9:57_3_1_2"/><b>then </b>
( <font color="Maroon" title="c2">r</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">k</font> or <font color="Maroon" title="c2">r</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E5:57_3_1_2"><i><font color="Green" title="E71">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E10:57_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E12:57"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></i>;<br/><a NAME="E11:57_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/><a NAME="E12:57_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">k1</font>
 <b>by </b><i><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/><a NAME="E13:57_3_1_2"/><b>then </b>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:57"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E14:57_3_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></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="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></i>;<br/><a NAME="E15:57_3_1_2"/><b>then </b><i><font color="Green" title="E72">A24</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>;<br/><a NAME="E16:57_3_1_2"/><b>then </b><i><font color="Green" title="E73">A25</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E17:57_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
 <b>by </b><i><a class="txt" href="nat_4.html#E15:57_3_1_2"><i><font color="Green" title="E72">A24</font></i></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></i>;<br/><a NAME="E18:57_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T48" title="XCMPLX_1:th.48">XCMPLX_1:48</a></i>;<br/><a NAME="E19:57_3_1_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E20:57_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E21:57_3_1_2"/><b>then </b>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E22:57_3_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <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">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E4:57_3"><i><font color="Green" title="E68">A13</font></i></a>, <a class="txt" href="nat_4.html#E16:57_3_1_2"><i><font color="Green" title="E73">A25</font></i></a>, <a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T110" title="RVSUM_1:th.110">RVSUM_1:110</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T110" title="RVSUM_1:th.110">RVSUM_1:110</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font>

;<br/>
<b>hence </b><a NAME="E17:57"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E15:57"><i><font color="Green" title="E65">A10</font></i></a>, <a class="ref" href="integra5.html#T3" title="INTEGRA5:th.3">INTEGRA5:3</a>, <a class="txt" href="nat_4.html#E14:57"><i><font color="Green" title="E64">A9</font></i></a></i>;<br/>


</div>
