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

<b>let </b><font color="Maroon">s</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<i><font color="Green">E41</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="newton.html#K11" target="_self">Newton_Coeff</a> <font color="Maroon">s</font></span>)</span> = 
<font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">1,1 <a href="newton.html#K10" target="_self">In_Power</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E2:48"/>
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="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K11" target="_self">Newton_Coeff</a> <font color="Maroon">s</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="newton.html#K11" target="_self">Newton_Coeff</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1,1 <a href="newton.html#K10" target="_self">In_Power</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">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">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="E1:48_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K11" target="_self">Newton_Coeff</a> <font color="Maroon">s</font></span>)</span>
 ;<br/>

<i><font color="Green">E45</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K11" target="_self">Newton_Coeff</a> <font color="Maroon">s</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="newton.html#K11" target="_self">Newton_Coeff</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/>
<b>then </b><i><font color="Green">E46</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K11" target="_self">Newton_Coeff</a> <font color="Maroon">s</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default">1,1 <a href="newton.html#K10" target="_self">In_Power</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default">1,1 <a href="newton.html#K10" target="_self">In_Power</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E4:48_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E6:48_2"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">l1</font> = <font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<span class="p1">(<span class="default">1,1 <a href="newton.html#K10" target="_self">In_Power</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="newton.html#K9" target="_self">choose</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="newton.html#K3" target="_self">|^</a> <font color="Maroon">l1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="newton.html#K3" target="_self">|^</a> <font color="Maroon">m1</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="newton.html#K9" target="_self">choose</a> <font color="Maroon">m1</font></span>)</span> <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="newton.html#K3" target="_self">|^</a> <font color="Maroon">m1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="newton.html#K9" target="_self">choose</a> <font color="Maroon">m1</font></span>)</span> <a href="nat_1.html#K2">*</a> 1
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="newton.html#K11" target="_self">Newton_Coeff</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E9:48_2"/>
<span class="p1">(<span class="default"><a href="newton.html#K11" target="_self">Newton_Coeff</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1,1 <a href="newton.html#K10" target="_self">In_Power</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:48"/>
 <a href="newton.html#K11" target="_self">Newton_Coeff</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 1,1 <a href="newton.html#K10" target="_self">In_Power</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/>


</div>
