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

<a NAME="E1:37"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  holds   <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Olive">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Olive">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ] <b>means </b> <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span>;<br/>
<a NAME="E1:37_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:37_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> is    <a href="matrix_1.html#M1">Matrix</a> of 0 <a href="binop_2.html#K23">+</a> 1,0, <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="matrix_1.html#T14">MATRIX_1:14</a></i>;<br/>
<a NAME="E2:37_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E3:37_1_1"/><b>then </b>
 <a href="matrix_1.html#K1">width</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/>
<a NAME="E4:37_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<b>thus </b> <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> = 
0
<b>by </b><i>, </i>
<br/>.= 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:37_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Olive">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:37_1_2"/><i><font color="Green">E58</font></i>: 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span>
 ;<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"><span class="p3"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrlin.html#K9">^^</a> <span class="p3"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p4"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrlin.html#K9">^^</a> <span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#D2">MATRLIN:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> 1

;<br/>
<b>then </b><i><font color="Green">E59</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrlin.html#K9">^^</a> <span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span></span>)</span> = 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2_3"><b>now </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="E1:37_1_2_3"/><i><font color="Green">E61</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"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span></span>)</span>
 ;<br/><a NAME="E2:37_1_2_3"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrlin.html#K9">^^</a> <span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&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="E3:37_1_2_3"/>
<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>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E4:37_1_2_3"/><b>then </b><i><font color="Green">E63</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/><b>reconsider </b><font color="Maroon">M1</font> = <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font>, <font color="Maroon">M2</font> = <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font> as   <a href="finseq_1.html#NM1">FinSequence</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrlin.html#K9">^^</a> <span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span></span>)</span> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">M1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">M2</font>
<b>by </b><i>, <a class="ref" href="matrlin.html#D2">MATRLIN:def 2</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">M2</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_2_4"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2_4_1"><b>suppose </b></a><a NAME="E1:37_1_2_4_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:37_1_2_4_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b> <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> = 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#T19">MATRLIN:19</a></i>
<br/>.= 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2_4_2"><b>suppose </b></a><a NAME="E1:37_1_2_4_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:37_1_2_4_2"/><b>then </b><i><font color="Green">E117</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:37_1_2_4_2"/><i><font color="Green">E118</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E4:37_1_2_4_2"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/><a NAME="E5:37_1_2_4_2"/><i><font color="Green">E120</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>then </b><i><font color="Green">E121</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span>
<b>by </b><i><a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><a NAME="E7:37_1_2_4_2"/><i><font color="Green">E122</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>then </b><i><font color="Green">E123</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</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="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&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">p</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/><i><font color="Green">E124</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span>
<b>by </b><i>, , <a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i>, , <a class="txt" href="matrprob.html#E1:37"><i><font color="Green">E43</font></i></a>, <a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>
;<br/><i><font color="Green">E125</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p4"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </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"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span></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">(<span class="default"><span class="p3"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p4"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</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="matrix_1.html#K1">width</a> <span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span></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">(<span class="default"><span class="p3"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p4"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="matrprob.html#E1:37"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
;<br/><a NAME="E11:37_1_2_4_2"/><i><font color="Green">E126</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/><a NAME="E12:37_1_2_4_2"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , , <a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">d1</font> = <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a>  as    <a href="matrix_1.html#M1">Matrix</a> of  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>,1, <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="txt" href="matrprob.html#E1:37_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/><a NAME="E14:37_1_2_4_2"/><i><font color="Green">E128</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="matrprob.html#E1:37"><i><font color="Green">E43</font></i></a>, <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/><a NAME="E15:37_1_2_4_2"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , , <a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, <a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">d2</font> = <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a>  as    <a href="matrix_1.html#M1">Matrix</a> of 1,1, <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/><a NAME="E17:37_1_2_4_2"/><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span>
 <b>by </b><i>, <a class="txt" href="matrprob.html#E1:37"><i><font color="Green">E43</font></i></a>, <a class="ref" href="matrix_2.html#T15">MATRIX_2:15</a></i>;<br/><i><font color="Green">E131</font></i>: <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">d2</font></span>)</span> <a href="matrix_1.html#K4">@</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrlin.html#K9">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p4"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, <a class="ref" href="matrlin.html#T32">MATRLIN:32</a></i>
<br/>.= 
<span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrlin.html#K9">^^</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span>
<b>by </b><i>, , , , <a class="ref" href="matrix_2.html#T15">MATRIX_2:15</a></i>
<br/>.= 
<span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span>
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> 
<b>by </b><i>, , <a class="ref" href="matrix_2.html#T15">MATRIX_2:15</a></i>
;<br/><b>thus </b> <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> = 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrlin.html#K9">^^</a> <span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p4"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#T19">MATRLIN:19</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">d1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">d2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">d2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="matrprob.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrprob.html#K2" target="_self">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="matrprob.html#E2:37_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="matrix_2.html#T11">MATRIX_2:11</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:37_1"/>
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>]
 
<b>from </b><i><a class="ref" href="finseq_2.html#S2">FINSEQ_2:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:37_1"/>
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="matrix_1.html#K4">@</a> </span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">M</font> being  <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a>   st  <a href="finseq_1.html#K3">len</a> <font color="Olive">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Olive">M</font> <a href="hidden.html#R1">=</a>  <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><font color="Olive">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>;<br/>
<a NAME="E2:37"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:37_2"/><i><font color="Green">E133</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:37_2"/><b>then </b>
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>;<br/>
<a NAME="E3:37_2"/><b>then </b><i><font color="Green">E134</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<b>thus </b> <a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Maroon">M</font> = 
0
<b>by </b><i>, </i>
<br/>.= 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:37"/><i><font color="Green">E135</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">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="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:37_3"/><i><font color="Green">E136</font></i>: 
for <font color="Olive">M</font> being  <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a>   st  <a href="finseq_1.html#K3">len</a> <font color="Olive">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> holds <br/> <a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Olive">M</font> <a href="hidden.html#R1">=</a>  <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><font color="Olive">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 ;<br/>

<b>thus </b><a NAME="E2:37_3"/>
for <font color="Olive">M</font> being  <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a>   st  <a href="finseq_1.html#K3">len</a> <font color="Olive">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 holds <br/> <a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Olive">M</font> <a href="hidden.html#R1">=</a>  <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><font color="Olive">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:37_3_1"/><i><font color="Green">E137</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<a NAME="E2:37_3_1"/><i><font color="Green">E138</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <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">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_3_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_1_1"><b>suppose </b></a><a NAME="E1:37_3_1_1_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">M</font> <a href="matrprob.html#K1" target="_self">.</a> 1 as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E3:37_3_1_1_1"/>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">g</font></span><a href="matrlin.html#K3">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E4:37_3_1_1_1"/>
 <a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_1_2"><b>suppose </b></a><a NAME="E1:37_3_1_1_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:37_3_1_1_2"/><i><font color="Green">E141</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K2">Del</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="matrlin.html#T3">MATRLIN:3</a></i>;<br/><a NAME="E3:37_3_1_1_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><a NAME="E4:37_3_1_1_2"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">M</font> <a href="matrprob.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<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="matrix_2.html#T18">MATRIX_2:18</a></i>;<br/><b>reconsider </b><font color="Maroon">M1</font> = <font color="Maroon">M</font> <a href="matrprob.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><b>reconsider </b><font color="Maroon">M'</font> = <font color="Maroon">M</font> as    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1, <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/><b>reconsider </b><font color="Maroon">r</font> = <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">M1</font></span><a href="matrlin.html#K3">*&gt;</a></span> as    <a href="matrix_1.html#M1">Matrix</a> of 1, <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><b>reconsider </b><font color="Maroon">w</font> =  <a href="matrlin.html#K2">Del</a> <font color="Maroon">M'</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> as    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">n</font>, <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="matrlin.html#T5">MATRLIN:5</a></i>;<br/><i><font color="Green">E145</font></i>:  <a href="matrix_1.html#K1">width</a> <font color="Maroon">w</font> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M'</font>
<b>by </b><i>, <a class="ref" href="matrlin.html#T4">MATRLIN:4</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="matrlin.html#T4">MATRLIN:4</a></i>
;<br/><i><font color="Green">E146</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="matrix_1.html#K4">@</a> </span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
;<br/><b>thus </b> <a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Maroon">M</font> = 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrlin.html#T6">MATRLIN:6</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">w</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p2">(<span class="default"><a href="matrlin.html#K2">Del</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrlin.html#K2">Del</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrlin.html#K2">Del</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrlin.html#K9">^^</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E1:37"><i><font color="Green">E43</font></i></a>, </i>
<br/>.= 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">r</font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="matrlin.html#T32">MATRLIN:32</a></i>
<br/>.= 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="matrlin.html#T6">MATRLIN:6</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<a NAME="E4:37"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">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#S2">NAT_1:sch 2</a>(, );<br/></i>
<a NAME="E5:37"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>]
 
;<br/>
<b>hence </b><a NAME="E6:37"/>
 <a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 ;<br/>


</div>
