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

<b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> , <font color="Maroon">g</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:61"/><i><font color="Green">E35</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> )
 ;<br/>

<b>set </b><font color="Maroon">n'</font> =  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>;<br/>
<a NAME="E2:61"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a>, <a class="ref" href="matrixc1.html#T4" target="_self">Th4</a></i>;<br/>
<b>set </b><font color="Maroon">q</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">p</font> = <font color="Maroon">g</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>;<br/>
<a NAME="E3:61"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:61"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&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> <font color="Maroon">g</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="matrixc1.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>
;<br/>
<b>then </b><i><font color="Green">E40</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> = 
 <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">g</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></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#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E7:61"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:61_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:61_3"/><b>then </b>
<font color="Maroon">u</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#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:61_3"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; 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="xcmplx_0.html#K2">+</a> 1 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_3_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_1_1_1"><b>case </b></a><a NAME="E1:61_3_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:61_3_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="matrixc1.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E3:61_3_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="matrixc1.html#T4" target="_self">Th4</a>, , , , <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font> = 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T4" target="_self">Th4</a>, , , <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_3_1_1_1"><i><font color="Green">E54</font></i></a>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_1_1_2"><b>case </b></a><a NAME="E1:61_3_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:61_3_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="matrixc1.html#E6:61_3_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:61_3_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:61_3_1_1_2"/><b>then </b>
<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#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="matrixc1.html#E6:61_3_1_1_2"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E5:61_3_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><a NAME="E6:61_3_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><a NAME="E7:61_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E8:61_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:61_3"/>
<font color="Maroon">f</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E9:61"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b> <a href="matrixc1.html#K6" target="_self">Sum</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <font color="Maroon">g</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <font color="Maroon">g</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/><br/>


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