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

<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">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default">(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c1">s</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <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"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c1">s</font></span>)</span></span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> $1</span>)</span>;<br/>
<a NAME="E1:43"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c3">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">n</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">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">G1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <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/>
<span class="kw">set </span><font color="Maroon" title="c8">G2</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E3:43_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E27">A2</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" target="_self" title="NEWTON:th.6">Th11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T87" title="RVSUM_1:th.87">RVSUM_1:87</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T87" title="RVSUM_1:th.87">RVSUM_1:87</a></span>
;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></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"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
<br/>.= 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T2" target="_self" title="NEWTON:th.2">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">F2</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T92" title="FINSEQ_2:th.92">FINSEQ_2:92</a></span>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G1</font> = 
<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="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
<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="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
<br/>.= 
<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="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T2" target="_self" title="NEWTON:th.2">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">F1</font> = <font color="Maroon" title="c7">G1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T92" title="FINSEQ_2:th.92">FINSEQ_2:92</a></span>;<br/>
<span class="lab"><font color="Green" title="E28">A3</font></span>:  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">F2</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T76" title="RVSUM_1:th.76">RVSUM_1:76</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>

;<br/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c10">F1</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T74" title="RVSUM_1:th.74">RVSUM_1:74</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>

;<br/>
<a NAME="E11:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A4</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p4">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="rvsum_1.html#T89" title="RVSUM_1:th.89">RVSUM_1:89</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">F</font> = <font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font>;<br/>
<a NAME="E12:43_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E13:43_1"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E14:43_1"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E15:43_1"/><span class="lab"><font color="Green" title="E33">A8</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">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</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">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></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="43_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">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="c12">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">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1_6"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<font color="Maroon" title="c12">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">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></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"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c13">r2</font> = <font color="Maroon" title="c9">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">r1</font> = <font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">r</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font>;<br/>
<a NAME="E2:43_1_6"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<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/>
<a NAME="E3:43_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A11</font></span>: 
<font color="Maroon" title="c12">i</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"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_6"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:43_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<font color="Maroon" title="c12">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> <font color="Maroon" title="c10">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:43_1_6"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
<font color="Maroon" title="c12">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> <font color="Maroon" title="c9">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E3:43_1_6"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:43_1_6"/><span class="lab"><font color="Green" title="E39">A14</font></span>: 
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_6_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:43_1_6_1"/>
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E2:43_1_6_1"/><span class="kw">then </span>
1 <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"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E3:43_1_6_1"/><span class="kw">then </span>
1 <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></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/>
<a NAME="E4:43_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A15</font></span>: 
1 <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="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</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/>
<a NAME="E5:43_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A16</font></span>: 
1 <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="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T3" target="_self" title="NEWTON:th.3">Th7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:43_1_6_1"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span><br/>

<a NAME="E7:43_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A17</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E8:43_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A18</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_6_1"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E9:43_1_6_1"/>
<span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_1_6_1"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="txt" href="#E4:43_1_6_1"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E10:43_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A19</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c6">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_1_6_1"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="ref" href="newton.html#T28" target="_self" title="NEWTON:th.28">Th39</a></span>;<br/>
<span class="lab"><font color="Green" title="E45">A20</font></span>: <font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_6"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="#E7:43_1_6_1"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_6_1"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c6">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_1_6_1"><span class="lab"><font color="Green" title="E43">A18</font></span></a>, <a class="txt" href="#E10:43_1_6_1"><span class="lab"><font color="Green" title="E44">A19</font></span></a>, <a class="ref" href="rvsum_1.html#T44" title="RVSUM_1:th.44">RVSUM_1:44</a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</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#T6" target="_self" title="NEWTON:th.6">Th11</a></span>
;<br/>
<a NAME="E12:43_1_6_1"/><span class="lab"><font color="Green" title="E46">A21</font></span>: 
<font color="Maroon" title="c9">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_6"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E13:43_1_6_1"/><span class="lab"><font color="Green" title="E47">A22</font></span>: 
<font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_6"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<font color="Maroon" title="c9">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_6"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E7:43_1_6_1"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1_6_1"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="txt" href="#E12:43_1_6_1"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="ref" href="rvsum_1.html#T11" title="RVSUM_1:th.11">RVSUM_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_1_6_1"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="txt" href="#E11:43_1_6_1"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="ref" href="newton.html#T28" target="_self" title="NEWTON:th.28">Th39</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:43_1_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">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 NAME="E7:43_1_6"/>
<font color="Maroon" title="c12">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="#E3:43_1_6"><span class="lab"><font color="Green" title="E36">A11</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="c16">j</font> = <font color="Maroon" title="c12">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/>
<span class="kw">set </span><font color="Maroon" title="c17">x</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font>;<br/>
<a NAME="E9:43_1_6"/><span class="lab"><font color="Green" title="E40">A23</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E10:43_1_6"/><span class="lab"><font color="Green" title="E41">A24</font></span>: 
<font color="Maroon" title="c12">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"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E3:43_1_6"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:43_1_6"/><span class="lab"><font color="Green" title="E42">A25</font></span>: 
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) </span> } </span>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_6_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_1_6_2"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">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>  : ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) </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"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span><br/>

<a NAME="E2:43_1_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A26</font></span>: 
 ex <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 <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">m1</font> = <font color="Maroon" title="c12">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="E4:43_1_6_2"/><span class="lab"><font color="Green" title="E44">A27</font></span>: 
<font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_6"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E5:43_1_6_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_1_6"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="txt" href="#E2:43_1_6_2"><span class="lab"><font color="Green" title="E43">A26</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">m2</font> = <font color="Maroon" title="c16">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="E7:43_1_6_2"/><span class="lab"><font color="Green" title="E45">A28</font></span>: 
<font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</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="#E9:43_1_6"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="txt" href="#E2:43_1_6_2"><span class="lab"><font color="Green" title="E43">A26</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E8:43_1_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</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
 
<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="c20">l2</font> = <font color="Maroon" title="c6">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c19">m2</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 NAME="E10:43_1_6_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_1_6"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="txt" href="#E2:43_1_6_2"><span class="lab"><font color="Green" title="E43">A26</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E11:43_1_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">j</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"><font color="Maroon" title="c6">n</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="txt" href="#E7:43_1_6_2"><span class="lab"><font color="Green" title="E45">A28</font></span></a></span>;<br/>
<a NAME="E12:43_1_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">j</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></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/>
<a NAME="E13:43_1_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A29</font></span>: 
<font color="Maroon" title="c16">j</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="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</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/>
<a NAME="E14:43_1_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A30</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E15:43_1_6_2"/><span class="lab"><font color="Green" title="E48">A31</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</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="#E2:43_1_6_2"><span class="lab"><font color="Green" title="E43">A26</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E16:43_1_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</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"><font color="Maroon" title="c6">n</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="txt" href="#E2:43_1_6_2"><span class="lab"><font color="Green" title="E43">A26</font></span></a></span>;<br/>
<a NAME="E17:43_1_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></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/>
<a NAME="E18:43_1_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A32</font></span>: 
<font color="Maroon" title="c12">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">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</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/>
<a NAME="E19:43_1_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A33</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E20:43_1_6_2"/>
<font color="Maroon" title="c12">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"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43_1_6_2"><span class="lab"><font color="Green" title="E49">A32</font></span></a>, <a class="ref" href="newton.html#T3" target="_self" title="NEWTON:th.3">Th7</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E51">A34</font></span>: <font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_6_2"><span class="lab"><font color="Green" title="E44">A27</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:43_1_6_2"><span class="lab"><font color="Green" title="E50">A33</font></span></a>, <a class="ref" href="rvsum_1.html#T44" title="RVSUM_1:th.44">RVSUM_1:44</a></span>
;<br/>
<a NAME="E22:43_1_6_2"/>
<font color="Maroon" title="c12">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:43_1_6_2"><span class="lab"><font color="Green" title="E48">A31</font></span></a>, <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="c21">l1</font> = <font color="Maroon" title="c6">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c18">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 NAME="E24:43_1_6_2"/><span class="lab"><font color="Green" title="E52">A35</font></span>: 
<font color="Maroon" title="c21">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c19">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E25:43_1_6_2"/><span class="lab"><font color="Green" title="E53">A36</font></span>: 
<font color="Maroon" title="c16">j</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="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1_6_2"><span class="lab"><font color="Green" title="E46">A29</font></span></a>, <a class="ref" href="newton.html#T3" target="_self" title="NEWTON:th.3">Th7</a></span>;<br/>
<a NAME="E26:43_1_6_2"/><span class="lab"><font color="Green" title="E54">A37</font></span>: 
<font color="Maroon" title="c9">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_6"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c9">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_1_6"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:43_1_6_2"><span class="lab"><font color="Green" title="E53">A36</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43_1_6_2"><span class="lab"><font color="Green" title="E47">A30</font></span></a>, <a class="ref" href="rvsum_1.html#T44" title="RVSUM_1:th.44">RVSUM_1:44</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_1_6"><span class="lab"><font color="Green" title="E41">A24</font></span></a>, <a class="txt" href="#E4:43_1_6_2"><span class="lab"><font color="Green" title="E44">A27</font></span></a>, <a class="txt" href="#E26:43_1_6_2"><span class="lab"><font color="Green" title="E54">A37</font></span></a>, <a class="txt" href="#E21:43_1_6_2"><span class="lab"><font color="Green" title="E51">A34</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c21">l1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c18">m1</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c18">m1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43_1_6_2"><span class="lab"><font color="Green" title="E49">A32</font></span></a>, <a class="txt" href="#E19:43_1_6_2"><span class="lab"><font color="Green" title="E50">A33</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="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c21">l1</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c18">m1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c18">m1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c18">m1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c18">m1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" target="_self" title="NEWTON:th.6">Th11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c18">m1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c18">m1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c19">m2</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c19">m2</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c20">l2</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1_6_2"><span class="lab"><font color="Green" title="E46">A29</font></span></a>, <a class="txt" href="#E14:43_1_6_2"><span class="lab"><font color="Green" title="E47">A30</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="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c18">m1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c18">m1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c19">m2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c19">m2</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c20">l2</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c19">m2</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c20">l2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" target="_self" title="NEWTON:th.6">Th11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c19">m2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T22" target="_self" title="NEWTON:th.22">Th32</a></span>
<br/>.= 
<span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_6"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E24:43_1_6_2"><span class="lab"><font color="Green" title="E52">A35</font></span></a>, <a class="ref" href="newton.html#D4" target="_self" title="NEWTON:def.4">Def4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E29:43_1_6_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">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 NAME="E12:43_1_6"/><span class="lab"><font color="Green" title="E43">A38</font></span>: 
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_6_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_1_6_3"/><span class="lab"><font color="Green" title="E44">A39</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span><br/>

<a NAME="E2:43_1_6_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A40</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E46">A41</font></span>: <font color="Maroon" title="c16">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_6_3"><span class="lab"><font color="Green" title="E44">A39</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1

;<br/>
<a NAME="E4:43_1_6_3"/>
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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"><font color="Maroon" title="c6">n</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="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></span>;<br/>
<a NAME="E5:43_1_6_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">j</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_6_3"><span class="lab"><font color="Green" title="E46">A41</font></span></a>, <a class="ref" href="newton.html#D4" target="_self" title="NEWTON:def.4">Def4</a></span>;<br/>
<a NAME="E6:43_1_6_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A42</font></span>: 
<font color="Maroon" title="c16">j</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="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</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="lab"><font color="Green" title="E48">A43</font></span>: <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font> = 
<span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</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="txt" href="#E3:43_1_6_3"><span class="lab"><font color="Green" title="E46">A41</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c6">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T29" target="_self" title="NEWTON:th.29">Th40</a></span>
;<br/>
<a NAME="E8:43_1_6_3"/><span class="lab"><font color="Green" title="E49">A44</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1_6_3"><span class="lab"><font color="Green" title="E47">A42</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E9:43_1_6_3"/><span class="lab"><font color="Green" title="E50">A45</font></span>: 
<font color="Maroon" title="c16">j</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="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1_6_3"><span class="lab"><font color="Green" title="E47">A42</font></span></a>, <a class="ref" href="newton.html#T3" target="_self" title="NEWTON:th.3">Th7</a></span>;<br/>
<span class="lab"><font color="Green" title="E51">A46</font></span>: <font color="Maroon" title="c9">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_6"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E2:43_1_6_3"><span class="lab"><font color="Green" title="E45">A40</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_6_3"><span class="lab"><font color="Green" title="E46">A41</font></span></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_1_6_3"><span class="lab"><font color="Green" title="E50">A45</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c6">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_1_6_3"><span class="lab"><font color="Green" title="E49">A44</font></span></a>, <a class="txt" href="#E7:43_1_6_3"><span class="lab"><font color="Green" title="E48">A43</font></span></a>, <a class="ref" href="rvsum_1.html#T44" title="RVSUM_1:th.44">RVSUM_1:44</a></span>
<br/>.= 
<font color="Maroon" title="c5">b</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</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#T6" target="_self" title="NEWTON:th.6">Th11</a></span>
;<br/>
<a NAME="E11:43_1_6_3"/><span class="lab"><font color="Green" title="E52">A47</font></span>: 
<font color="Maroon" title="c9">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_6"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</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/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T2" target="_self" title="NEWTON:th.2">Th6</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E53">A48</font></span>: <font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p4">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_6"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="#E2:43_1_6_3"><span class="lab"><font color="Green" title="E45">A40</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T42" title="FINSEQ_1:th.42">FINSEQ_1:42</a></span>
;<br/>
<a NAME="E14:43_1_6_3"/>
<font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_6"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1_6_3"><span class="lab"><font color="Green" title="E53">A48</font></span></a>, <a class="txt" href="#E11:43_1_6_3"><span class="lab"><font color="Green" title="E52">A47</font></span></a>, <a class="ref" href="rvsum_1.html#T11" title="RVSUM_1:th.11">RVSUM_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_6_3"><span class="lab"><font color="Green" title="E45">A40</font></span></a>, <a class="txt" href="#E10:43_1_6_3"><span class="lab"><font color="Green" title="E51">A46</font></span></a>, <a class="ref" href="newton.html#T29" target="_self" title="NEWTON:th.29">Th40</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:43_1_6_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">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>
<div><span class="lab"><font color="Green" title="E44">A49</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_6_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_1_6_4"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"> { <span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">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>  : ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) </span> } </span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span><br/><a NAME="E2:43_1_6_4"/><span class="kw">then </span>
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">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>  : ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) </span> } </span>  or <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:43_1_6_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1_6"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="txt" href="#E11:43_1_6"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="txt" href="#E12:43_1_6"><span class="lab"><font color="Green" title="E43">A38</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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><span class="kw">end;</span></div>
<a NAME="E14:43_1_6"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_6"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:43_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_6"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E13:43_1_6"><span class="lab"><font color="Green" title="E44">A49</font></span></a>, <a class="ref" href="newton.html#T1" target="_self" title="NEWTON:th.1">Th5</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</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><span class="kw">end;</span></div>
<a NAME="E16:43_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c9">F2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="newton.html#D4" target="_self" title="NEWTON:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:43_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</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="#E4:43_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E11:43_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E15:43_1"><span class="lab"><font color="Green" title="E33">A8</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><span class="kw">end;</span></div>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T11" title="FINSOP_1:th.11">FINSOP_1:11</a>, <a class="ref" href="rvsum_1.html#T94" title="RVSUM_1:th.94">RVSUM_1:94</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T27" target="_self" title="NEWTON:th.27">Th38</a></span>
;<br/>
<a NAME="E3:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A50</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E4:43"/>
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E27">A50</font></span></a>, <a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E26">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:43"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="newton.html#K9" title="NEWTON:func.9">In_Power</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"> verum</span><br/>


</div>
