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<span class="kw">let </span><font color="Maroon" title="c1">s</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="newton.html#K10" title="NEWTON:func.10">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#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c1">s</font></span><br/>

<a NAME="E1:45"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 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#K10" title="NEWTON:func.10">Newton_Coeff</a> <font color="Maroon" title="c1">s</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="newton.html#K10" title="NEWTON:func.10">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#K9" title="NEWTON:func.9">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="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K10" title="NEWTON:func.10">Newton_Coeff</a> <font color="Maroon" title="c1">s</font></span>)</span> implies <span class="p1">(<span class="default"><a href="newton.html#K10" title="NEWTON:func.10">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#K9" title="NEWTON:func.9">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><br/>

<span class="kw">assume </span><a NAME="E1:45_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<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#K10" title="NEWTON:func.10">Newton_Coeff</a> <font color="Maroon" title="c1">s</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="newton.html#K10" title="NEWTON:func.10">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#K9" title="NEWTON:func.9">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><br/>

<span class="lab"><font color="Green" title="E30">A3</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K10" title="NEWTON:func.10">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#K10" title="NEWTON:func.10">Newton_Coeff</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#D5" target="_self" title="NEWTON:def.5">Def5</a></span>
;<br/>
<a NAME="E3:45_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/>
<a NAME="E5:45_1"/>
<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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E6:45_1"/><span class="kw">then </span>
<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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K10" title="NEWTON:func.10">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#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="newton.html#D4" target="_self" title="NEWTON:def.4">Def4</a></span>
<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#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c1">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default">(1,1) <a href="newton.html#K9" title="NEWTON:func.9">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#K8" title="NEWTON:func.8">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="newton.html#D4" target="_self" title="NEWTON:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="newton.html#K8" title="NEWTON:func.8">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" target="_self" title="NEWTON:th.10">Th15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="newton.html#K8" title="NEWTON:func.8">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 class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" target="_self" title="NEWTON:th.10">Th15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="newton.html#K10" title="NEWTON:func.10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="newton.html#D5" target="_self" title="NEWTON:def.5">Def5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:45_1"/>
<span class="p1">(<span class="default"><a href="newton.html#K10" title="NEWTON:func.10">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#K9" title="NEWTON:func.9">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="newton.html#K10" title="NEWTON:func.10">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
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#D5" target="_self" title="NEWTON:def.5">Def5</a></span>
<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#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c1">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#D4" target="_self" title="NEWTON:def.4">Def4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:45"/>
 <a href="newton.html#K10" title="NEWTON:func.10">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#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c1">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="newton.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
