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

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

<b>assume </b><a NAME="E1:5"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 ;<br/>

<b>assume </b><a NAME="E2:5"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<a NAME="E3:5"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E4:5"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>set </b><font color="Maroon">f2a</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">f2b</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">f1</font> =  <a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">f2</font> = <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E5:5"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#D5">NEWTON:def 5</a></i>;<br/>
<i><font color="Green">E48</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i>, <a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1

;<br/>
<a NAME="E7:5"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#D5">NEWTON:def 5</a></i>;<br/>
<a NAME="E8:5"/><i><font color="Green">E54</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:5"/><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:5"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:5_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> or  ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">j</font> ) )
 </span><b>by </b><i>, , <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_1_1"><b>suppose </b></a><a NAME="E1:5_2_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:5_2_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_2_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> or  ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">j</font> ) )
 </span><b>by </b><i>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:5_2_1_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><div class="add"><a NAME="E2:5_2_1_1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E3:5_2_1_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E4:5_2_1_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>set </b><font color="Maroon">k</font> = 1 <a href="binop_2.html#K10">-</a> 1;<br/><a NAME="E5:5_2_1_1_1_1"/><i><font color="Green">E64</font></i>: 
1 <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> 0
 ;<br/><b>reconsider </b><font color="Maroon">k</font> = 1 <a href="binop_2.html#K10">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="newton.html#D5">NEWTON:def 5</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="newton.html#T29">NEWTON:29</a></i>
;<br/><b>hence </b><a NAME="E8:5_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:5_2_1_1_1_2"/><i><font color="Green">E66</font></i>: 
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">j</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:5_2_1_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:5_2_1_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/><a NAME="E5:5_2_1_1_1_2"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E6:5_2_1_1_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T68">FINSEQ_2:68</a></i>;<br/><a NAME="E7:5_2_1_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>;<br/><a NAME="E8:5_2_1_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:5_2_1_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E10:5_2_1_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E11:5_2_1_1_1_2"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>set </b><font color="Maroon">k</font> = <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E12:5_2_1_1_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E13:5_2_1_1_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="newton.html#D5">NEWTON:def 5</a></i>;<br/><a NAME="E14:5_2_1_1_1_2"/>
<font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E15:5_2_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, </i>;<br/><b>hence </b><a NAME="E16:5_2_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_1_2"><b>suppose </b></a><a NAME="E1:5_2_1_2"/><i><font color="Green">E79</font></i>: 
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">j</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:5_2_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:5_2_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/><a NAME="E5:5_2_1_2"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E6:5_2_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E7:5_2_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><i><font color="Green">E85</font></i>: <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i>, <a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1

;<br/><b>reconsider </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="newton.html#D5">NEWTON:def 5</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="newton.html#T31">NEWTON:31</a></i>
;<br/><b>hence </b><a NAME="E11:5_2_1_2"/>
<span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:5"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="newton.html#K11">Newton_Coeff</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra5.html#T3">INTEGRA5:3</a>, </i>;<br/>
<a NAME="E12:5"/><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T77">FINSEQ_2:77</a></i>;<br/>
<a NAME="E13:5"/><i><font color="Green">E88</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E14:5"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:5"/>
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 1
 
<b>by </b><i>, , <a class="ref" href="newton.html#T33">NEWTON:33</a></i>;<br/>
<a NAME="E16:5"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E17:5"/><b>then </b><i><font color="Green">E91</font></i>: 
2 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E18:5"/><i><font color="Green">E92</font></i>: 
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T110">RVSUM_1:110</a></i>;<br/>
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T106">RVSUM_1:106</a></i>
<br/>.= 
2 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
2 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<a NAME="E20:5"/><b>then </b>
2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T44">NEWTON:44</a>, </i>;<br/>
<a NAME="E21:5"/><b>then </b>
2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E22:5"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/>
<b>hence </b><a NAME="E23:5"/>
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>


</div>
