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

<b>let </b><font color="Maroon" title="c1">s</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<i><font color="Green" title="E29">A1</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="newton.html#K11" title="NEWTON:func.11">Newton_Coeff</a> <font color="Maroon" title="c1">s</font></span>)</span> = 
<font color="Maroon" title="c1">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="newton.html#D5" target="_self" title="NEWTON:def.5">Def5</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">1,1 <a href="newton.html#K10" title="NEWTON:func.10">In_Power</a> <font color="Maroon" title="c1">s</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#D4" target="_self" title="NEWTON:def.4">Def4</a></i>
;<br/>
<a NAME="E2:48"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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"><a href="newton.html#K11" title="NEWTON:func.11">Newton_Coeff</a> <font color="Maroon" title="c1">s</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="newton.html#K11" title="NEWTON:func.11">Newton_Coeff</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1,1 <a href="newton.html#K10" title="NEWTON:func.10">In_Power</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c2">i</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"><a href="newton.html#K11" title="NEWTON:func.11">Newton_Coeff</a> <font color="Maroon" title="c1">s</font></span>)</span>
 ;<br/>

<i><font color="Green" title="E31">A3</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K11" title="NEWTON:func.11">Newton_Coeff</a> <font color="Maroon" title="c1">s</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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"><a href="newton.html#K11" title="NEWTON:func.11">Newton_Coeff</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#D5" target="_self" title="NEWTON:def.5">Def5</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E32">A4</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K11" title="NEWTON:func.11">Newton_Coeff</a> <font color="Maroon" title="c1">s</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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">1,1 <a href="newton.html#K10" title="NEWTON:func.10">In_Power</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#D4" target="_self" title="NEWTON:def.4">Def4</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default">1,1 <a href="newton.html#K10" title="NEWTON:func.10">In_Power</a> <font color="Maroon" title="c1">s</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E4:48_2"/><i><font color="Green" title="E33">A5</font></i>: 
( <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="newton.html#E1:48_2"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="newton.html#E2:48_2"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">m1</font> = <font color="Maroon" title="c2">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 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="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<a NAME="E6:48_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="txt" href="newton.html#E4:48_2"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">l1</font> = <font color="Maroon" title="c1">s</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">m1</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="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<span class="p1">(<span class="default">1,1 <a href="newton.html#K10" title="NEWTON:func.10">In_Power</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">l1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">m1</font></span>)</span>
<b>by </b><i><a class="txt" href="newton.html#E1:48_2"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="newton.html#E3:48_2"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="newton.html#D4" target="_self" title="NEWTON:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">m1</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T15" target="_self" title="NEWTON:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
<b>by </b><i><a class="ref" href="newton.html#T15" target="_self" title="NEWTON:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="newton.html#K11" title="NEWTON:func.11">Newton_Coeff</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
<b>by </b><i><a class="txt" href="newton.html#E1:48_2"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="newton.html#D5" target="_self" title="NEWTON:def.5">Def5</a></i>
;<br/>
<b>hence </b><a NAME="E9:48_2"/>
<span class="p1">(<span class="default"><a href="newton.html#K11" title="NEWTON:func.11">Newton_Coeff</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1,1 <a href="newton.html#K10" title="NEWTON:func.10">In_Power</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:48"/>
 <a href="newton.html#K11" title="NEWTON:func.11">Newton_Coeff</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1,1 <a href="newton.html#K10" title="NEWTON:func.10">In_Power</a> <font color="Maroon" title="c1">s</font>
 <b>by </b><i><a class="txt" href="newton.html#E1:48"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="newton.html#T3" target="_self" title="NEWTON:th.3">Th3</a></i>;<br/>


</div>
