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

<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="algstr_1.html#V1">left_zeroed</a> <a href="binom.html#V1" target="_self">add-cancelable</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:38"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <a href="binom.html#K2" target="_self">|^</a> 0 = 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E1:38"><i><font color="Green">E25</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:38"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:38_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>then </b><i><font color="Green">E41</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <a href="binom.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D11">VECTSP_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>set </b><font color="Maroon">G1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">G2</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span>;<br/>
<i><font color="Green">E44</font></i>:  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D3">POLYNOM1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green">E47</font></i>:  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D3">POLYNOM1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i/>
;<br/>
<b>then reconsider </b><font color="Maroon">F1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i/>
;<br/>
<b>then reconsider </b><font color="Maroon">F2</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E9:38_2"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E10:38_2"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<i><font color="Green">E54</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F2</font>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green">E55</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F1</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F2</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="algstr_1.html#D5">ALGSTR_1:def 5</a></i>
;<br/>
<a NAME="E14:38_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="binom.html#K1" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">F</font> = <font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font>;<br/>
<i><font color="Green">E57</font></i>:  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E16:38_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/>
<a NAME="E17:38_2"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_10"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:38_2_10"/><i><font color="Green">E61</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:38_2_10"/><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E3:38_2_10"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:38_2_10"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="binom.html#E2:38_2"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:38_2_10"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="binom.html#E2:38_2"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="binom.html#E2:38_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E7:38_2_10"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:38_2_10"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:38_2_10"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:38_2_10"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E11:38_2_10"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<b>set </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>;<br/>
<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">r1</font> = <font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">r2</font> = <font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>;<br/>
<a NAME="E12:38_2_10"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="binom.html#E2:38_2"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E13:38_2_10"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> implies <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2_10_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:38_2_10_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:38_2_10_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E4:38_2_10_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:38_2_10_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:38_2_10_1"/><b>then </b><i><font color="Green">E81</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:38_2_10_1"/><b>then </b><i><font color="Green">E82</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:38_2_10_1"/><i><font color="Green">E83</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="binom.html#E11:38_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:38_2_10_1"/><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="txt" href="binom.html#E12:38_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<i><font color="Green">E86</font></i>: <font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="binom.html#E4:38_2"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="binom.html#E14:38_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="binom.html#E14:38_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="binom.html#E12:38_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="polynom1.html#D3">POLYNOM1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="txt" href="binom.html#E15:38_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="binom.html#E2:38"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/>
<i><font color="Green">E87</font></i>: <font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="txt" href="binom.html#E2:38_2"><i><font color="Green">E41</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="binom.html#E16:38_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="txt" href="binom.html#E2:38"><i><font color="Green">E26</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E14:38_2_10"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> implies <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_10_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2_10_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:38_2_10_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:38_2_10_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:38_2_10_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="binom.html#E2:38_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="binom.html#E12:38_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<i><font color="Green">E93</font></i>: <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="txt" href="binom.html#E11:38_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1

;<br/>
<i><font color="Green">E94</font></i>: <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>
;<br/>
<a NAME="E7:38_2_10_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E8:38_2_10_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="binom.html#E15:38_2"><i><font color="Green">E57</font></i></a>, </i>;<br/>
<a NAME="E9:38_2_10_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:38_2_10_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="binom.html#E16:38_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E98</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E15:38_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="binom.html#E17:38_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="binom.html#T1" target="_self">Th1</a></i>
;<br/>
<i><font color="Green">E100</font></i>: <font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E4:38_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="binom.html#E12:38_2"><i><font color="Green">E55</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/>
<i><font color="Green">E101</font></i>: <font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="binom.html#E12:38_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E15:38_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="binom.html#E1:38_2_10"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="binom.html#E1:38_2_10"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="binom.html#E17:38_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="binom.html#E2:38_2_10"><i><font color="Green">E62</font></i></a>, <a class="ref" href="polynom1.html#D3">POLYNOM1:def 3</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="binom.html#E14:38_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="txt" href="binom.html#E2:38_2"><i><font color="Green">E41</font></i></a>, , <a class="txt" href="binom.html#E3:38_2_10"><i><font color="Green">E63</font></i></a>, </i>
<br/>.= 
<font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E4:38_2_10"><i><font color="Green">E64</font></i></a>, <a class="ref" href="algstr_1.html#D5">ALGSTR_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="binom.html#E12:38_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="binom.html#T1" target="_self">Th1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E15:38_2_10"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 ) </span>}</span>  implies <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_10_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2_10_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 ) </span>}</span> 
 ;<br/>

<a NAME="E2:38_2_10_3"/><b>then </b><i><font color="Green">E103</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
;<br/>
<a NAME="E3:38_2_10_3"/><b>then </b><i><font color="Green">E104</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:38_2_10_3"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:38_2_10_3"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:38_2_10_3"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:38_2_10_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="binom.html#E15:38_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="binom.html#E12:38_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E9:38_2_10_3"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="binom.html#E14:38_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">l1</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E11:38_2_10_3"/><i><font color="Green">E110</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="binom.html#E9:38_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="binom.html#E12:38_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:38_2_10_3"/><i><font color="Green">E111</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="binom.html#E9:38_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="binom.html#E12:38_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E13:38_2_10_3"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:38_2_10_3"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:38_2_10_3"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E16:38_2_10_3"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="binom.html#E2:38_2_10"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m2</font> = <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="binom.html#E17:38_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E18:38_2_10_3"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E19:38_2_10_3"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="binom.html#E1:38_2_10"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">l2</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E21:38_2_10_3"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="binom.html#E2:38_2_10"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:38_2_10_3"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="binom.html#E4:38_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E23:38_2_10_3"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<i><font color="Green">E163</font></i>: <font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="binom.html#E16:38_2"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">a</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="binom.html#E16:38_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="polynom1.html#D3">POLYNOM1:def 3</a></i>
;<br/>
<i><font color="Green">E164</font></i>: <font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E9:38_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="binom.html#E7:38_2_10"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="binom.html#E4:38_2_10"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K7">*</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="binom.html#E3:38_2_10"><i><font color="Green">E63</font></i></a>, <a class="ref" href="polynom1.html#D3">POLYNOM1:def 3</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="txt" href="binom.html#E2:38_2"><i><font color="Green">E41</font></i></a>, , <a class="txt" href="binom.html#E9:38_2_10"><i><font color="Green">E70</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m1</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">l1</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="txt" href="binom.html#E8:38_2_10"><i><font color="Green">E69</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">l1</font></span>)</span> <a href="binom.html#K6" target="_self">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="binom.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">l1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m1</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">m1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">l1</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m1</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">m1</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m1</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">m1</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m1</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">m1</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">m2</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m2</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">l2</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E3:38_2_10"><i><font color="Green">E63</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m1</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">m1</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">m2</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m2</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">l2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <span class="p4">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p4">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m2</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">l2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p4">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binom.html#K6" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <span class="p3">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m2</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">l2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <span class="p4">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m2</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">l2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binom.html#K6" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <span class="p4">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m2</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">l2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binom.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binom.html#K6" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <span class="p4">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m2</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">l2</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <span class="p4">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m2</font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binom.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <span class="p4">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binom.html#K7" target="_self">choose</a> <span class="p3">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binom.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T32">NEWTON:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="binom.html#E3:38_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="binom.html#E5:38_2_10"><i><font color="Green">E65</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="binom.html#E3:38_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_10_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_2_10_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:38_2_10_4"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 ) </span>}</span>  or <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:38_2_10_4"/>
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="binom.html#E10:38_2"><i><font color="Green">E52</font></i></a>, , <a class="txt" href="binom.html#E11:38_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:38_2_10"/>
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="binom.html#E3:38_2"><i><font color="Green">E44</font></i></a>, , <a class="ref" href="newton.html#T5">NEWTON:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:38_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="binom.html#K1" target="_self">+</a> <font color="Maroon">F2</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E19:38_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:38"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="binom.html#T2" target="_self">Th2</a>, );<br/></i>
<b>hence </b><a NAME="E4:38"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="binom.html#K8" target="_self">In_Power</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>


</div>
