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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>consider </b><font color="Maroon" title="c3">m1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:72"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">n1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:72"/><i><font color="Green" title="E60">A2</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">m1</font> = <font color="Maroon" title="c3">m1</font>, <font color="Maroon" title="c6">n1</font> = <font color="Maroon" title="c4">n1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c2">m</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">e1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E8:72"/><i><font color="Green" title="E61">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">m</font> holds <br/>( ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">m1</font> implies <font color="Maroon" title="c9">e1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( not <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">m1</font> implies <font color="Maroon" title="c9">e1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) ) )
 <b>by </b><i><a class="ref" href="matrprob.html#T2" target="_self" title="MATRPROB:th.2">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">e2</font> = <font color="Maroon" title="c8">m</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E10:72"/><i><font color="Green" title="E62">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">m</font> holds <br/><font color="Maroon" title="c10">e2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">M1</font> being    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c8">m</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E12:72"/><i><font color="Green" title="E63">A5</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font> holds <br/>( ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font> implies <font color="Maroon" title="c11">M1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e2</font> ) &amp; ( not <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font> implies <font color="Maroon" title="c11">M1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e1</font> ) )
 <b>by </b><i><a class="txt" href="matrprob.html#E8:72"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="matrprob.html#T19" target="_self" title="MATRPROB:th.19">Th19</a></i>;<br/>
<a NAME="E13:72"/><i><font color="Green" title="E64">A6</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">m</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">m</font> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E8:72"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="matrprob.html#E10:72"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green" title="E65">A7</font></i>:  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c10">e2</font> = 
<font color="Maroon" title="c8">m</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="rvsum_1.html#T110" title="RVSUM_1:th.110">RVSUM_1:110</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
<a NAME="E15:72"/><i><font color="Green" title="E66">A8</font></i>: 
<font color="Maroon" title="c5">m1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E16:72"/><i><font color="Green" title="E67">A9</font></i>: 
( <font color="Maroon" title="c5">m1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">m1</font> &amp; <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">m1</font> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E17:72"/><b>then </b>
not <font color="Maroon" title="c5">m1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">m1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E18:72"/><b>then </b>
( <font color="Maroon" title="c9">e1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="matrprob.html#E8:72"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="matrprob.html#E15:72"><i><font color="Green" title="E66">A8</font></i></a></i>;<br/>
<a NAME="E19:72"/><b>then </b><i><font color="Green" title="E68">A10</font></i>: 
<font color="Maroon" title="c9">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">e1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">m1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T20" title="RFINSEQ:th.20">RFINSEQ:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">e3</font> = <font color="Maroon" title="c9">e1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">m1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E21:72"/><i><font color="Green" title="E69">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">e3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m1</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E8:72"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="matrprob.html#E16:72"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E22:72"/><i><font color="Green" title="E70">A12</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:72_2_1"/>
( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_2_1_1"><b>suppose </b></a><a NAME="E1:72_2_1_1"/>
<font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:72_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">e3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="matrprob.html#E8:72"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><a NAME="E3:72_2_1_1"/><b>then </b><i><font color="Green" title="E71">A13</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c12">e3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a></i>;<br/><b>thus </b> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">e1</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">e1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">m1</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
<b>by </b><i><a class="txt" href="matrprob.html#E19:72"><i><font color="Green" title="E68">A10</font></i></a>, <a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="matrprob.html#E3:72_2_1_1"><i><font color="Green" title="E71">A13</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_2_1_2"><b>suppose </b></a><a NAME="E1:72_2_1_2"/><i><font color="Green" title="E71">A14</font></i>: 
<font color="Maroon" title="c5">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:72_2_1_2"/><b>then </b><i><font color="Green" title="E72">A15</font></i>: 
<font color="Maroon" title="c5">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:72_2_1_2"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">e3</font> holds <br/><font color="Maroon" title="c12">e3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:72_2_1_2_1"/><i><font color="Green" title="E73">A16</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">e3</font>
 ;<br/>

<a NAME="E2:72_2_1_2_1"/><i><font color="Green" title="E74">A17</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">m1</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E21:72"><i><font color="Green" title="E69">A11</font></i></a>, <a class="txt" href="matrprob.html#E1:72_2_1_2_1"><i><font color="Green" title="E73">A16</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:72_2_1_2_1"/>
<font color="Maroon" title="c5">m1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E4:72_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="matrprob.html#E2:72_2_1_2"><i><font color="Green" title="E72">A15</font></i></a>, <a class="ref" href="finseq_1.html#T82" title="FINSEQ_1:th.82">FINSEQ_1:82</a></i>;<br/>
<a NAME="E5:72_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">e1</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E8:72"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:72_2_1_2_1"/><b>then </b>
( <font color="Maroon" title="c12">e3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">e1</font> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:72_2_1_2_1"><i><font color="Green" title="E74">A17</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/>
<b>hence </b><a NAME="E7:72_2_1_2_1"/>
<font color="Maroon" title="c12">e3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="matrprob.html#E8:72"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="matrprob.html#E13:72"><i><font color="Green" title="E64">A6</font></i></a>, <a class="txt" href="matrprob.html#E2:72_2_1_2_1"><i><font color="Green" title="E74">A17</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:72_2_1_2"/><b>then </b>
<font color="Maroon" title="c12">e3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m1</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="matrprob.html#E21:72"><i><font color="Green" title="E69">A11</font></i></a>, <a class="txt" href="matrprob.html#E1:72_2_1_2"><i><font color="Green" title="E71">A14</font></i></a>, <a class="ref" href="matrprob.html#T1" target="_self" title="MATRPROB:th.1">Th1</a></i>;<br/><b>then </b><i><font color="Green" title="E73">A18</font></i>:  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c12">e3</font> = 
<font color="Maroon" title="c5">m1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="rvsum_1.html#T110" title="RVSUM_1:th.110">RVSUM_1:110</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><b>thus </b> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">e1</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">e1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">m1</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
<b>by </b><i><a class="txt" href="matrprob.html#E19:72"><i><font color="Green" title="E68">A10</font></i></a>, <a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="matrprob.html#E5:72_2_1_2"><i><font color="Green" title="E73">A18</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E23:72"/><i><font color="Green" title="E71">A19</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c11">M1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font> holds <br/>( ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font> implies <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c11">M1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( not <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font> implies <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c11">M1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3"><b>proof </b></a><div class="add">

<b>thus </b><i><font color="Green" title="E72">A20</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c11">M1</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">M1</font>
<b>by </b><i><a class="ref" href="matrprob.html#D1" target="_self" title="MATRPROB:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>
;<br/>

<b>let </b><font color="Maroon" title="c13">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:72_3"/><i><font color="Green" title="E73">A21</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 ;<br/>

<a NAME="E3:72_3"/><i><font color="Green" title="E74">A22</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c11">M1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:72_3"><i><font color="Green" title="E72">A20</font></i></a>, <a class="txt" href="matrprob.html#E2:72_3"><i><font color="Green" title="E73">A21</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>thus </b><a NAME="E4:72_3"/>
( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font> implies <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c11">M1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_3_2"/><i><font color="Green" title="E75">A23</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c11">M1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">i</font> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">M1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E3:72_3"><i><font color="Green" title="E74">A22</font></i></a>, <a class="ref" href="matrprob.html#D1" target="_self" title="MATRPROB:def.1">Def1</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="matrprob.html#E12:72"><i><font color="Green" title="E63">A5</font></i></a>, <a class="txt" href="matrprob.html#E14:72"><i><font color="Green" title="E65">A7</font></i></a>, <a class="txt" href="matrprob.html#E2:72_3"><i><font color="Green" title="E73">A21</font></i></a>, <a class="txt" href="matrprob.html#E1:72_3_2"><i><font color="Green" title="E75">A23</font></i></a></i>
;<br/>


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

<b>assume </b><a NAME="E5:72_3"/><i><font color="Green" title="E75">A24</font></i>: 
not <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c11">M1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">i</font> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">M1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E3:72_3"><i><font color="Green" title="E74">A22</font></i></a>, <a class="ref" href="matrprob.html#D1" target="_self" title="MATRPROB:def.1">Def1</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="matrprob.html#E12:72"><i><font color="Green" title="E63">A5</font></i></a>, <a class="txt" href="matrprob.html#E22:72"><i><font color="Green" title="E70">A12</font></i></a>, <a class="txt" href="matrprob.html#E2:72_3"><i><font color="Green" title="E73">A21</font></i></a>, <a class="txt" href="matrprob.html#E5:72_3"><i><font color="Green" title="E75">A24</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E24:72"/><i><font color="Green" title="E72">A25</font></i>: 
 <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c11">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c13">e4</font> =  <a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c11">M1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E2:72_4"/><i><font color="Green" title="E73">A26</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">e4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E23:72"><i><font color="Green" title="E71">A19</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:72_4"/><i><font color="Green" title="E74">A27</font></i>: 
<font color="Maroon" title="c6">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E4:72"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E4:72_4"/><i><font color="Green" title="E75">A28</font></i>: 
( <font color="Maroon" title="c6">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">n1</font> &amp; <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">n1</font> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E4:72"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:72_4"/><b>then </b>
not <font color="Maroon" title="c6">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:72_4"/><b>then </b><i><font color="Green" title="E76">A29</font></i>: 
( <font color="Maroon" title="c13">e4</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">e4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="matrprob.html#E4:72"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="matrprob.html#E23:72"><i><font color="Green" title="E71">A19</font></i></a>, <a class="txt" href="matrprob.html#E3:72_4"><i><font color="Green" title="E74">A27</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">e5</font> = <font color="Maroon" title="c13">e4</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E8:72_4"/><i><font color="Green" title="E77">A30</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">e5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n1</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E23:72"><i><font color="Green" title="E71">A19</font></i></a>, <a class="txt" href="matrprob.html#E4:72_4"><i><font color="Green" title="E75">A28</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E9:72_4"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c13">e4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:72_4_1_1"/>
( <font color="Maroon" title="c6">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c6">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_4_1_1_1"><b>suppose </b></a><a NAME="E1:72_4_1_1_1"/>
<font color="Maroon" title="c6">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:72_4_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">e5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="matrprob.html#E23:72"><i><font color="Green" title="E71">A19</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><a NAME="E3:72_4_1_1_1"/><b>then </b><i><font color="Green" title="E78">A31</font></i>: 
<font color="Maroon" title="c14">e5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></i>;<br/><b>thus </b> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c13">e4</font> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">e4</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E6:72_4"><i><font color="Green" title="E76">A29</font></i></a>, <a class="ref" href="rfinseq.html#T20" title="RFINSEQ:th.20">RFINSEQ:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">e4</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n1</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
<b>by </b><i><a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="matrprob.html#E3:72_4_1_1_1"><i><font color="Green" title="E78">A31</font></i></a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_4_1_1_2"><b>suppose </b></a><a NAME="E1:72_4_1_1_2"/><i><font color="Green" title="E78">A32</font></i>: 
<font color="Maroon" title="c6">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:72_4_1_1_2"/><b>then </b><i><font color="Green" title="E79">A33</font></i>: 
<font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:72_4_1_1_2"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">e5</font> holds <br/><font color="Maroon" title="c14">e5</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:72_4_1_1_2_1"/><i><font color="Green" title="E80">A34</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">e5</font>
 ;<br/>

<a NAME="E2:72_4_1_1_2_1"/><i><font color="Green" title="E81">A35</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E8:72_4"><i><font color="Green" title="E77">A30</font></i></a>, <a class="txt" href="matrprob.html#E1:72_4_1_1_2_1"><i><font color="Green" title="E80">A34</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:72_4_1_1_2_1"/>
<font color="Maroon" title="c6">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E4:72_4_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E4:72"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="matrprob.html#E2:72_4_1_1_2"><i><font color="Green" title="E79">A33</font></i></a>, <a class="ref" href="finseq_1.html#T82" title="FINSEQ_1:th.82">FINSEQ_1:82</a></i>;<br/>
<a NAME="E5:72_4_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">e4</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E23:72"><i><font color="Green" title="E71">A19</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:72_4_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c14">e5</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">e4</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">e4</font> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:72_4_1_1_2_1"><i><font color="Green" title="E81">A35</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/>
<b>hence </b><a NAME="E7:72_4_1_1_2_1"/>
<font color="Maroon" title="c14">e5</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="matrprob.html#E23:72"><i><font color="Green" title="E71">A19</font></i></a>, <a class="txt" href="matrprob.html#E2:72_4"><i><font color="Green" title="E73">A26</font></i></a>, <a class="txt" href="matrprob.html#E2:72_4_1_1_2_1"><i><font color="Green" title="E81">A35</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:72_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">e5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n1</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="matrprob.html#E8:72_4"><i><font color="Green" title="E77">A30</font></i></a>, <a class="txt" href="matrprob.html#E1:72_4_1_1_2"><i><font color="Green" title="E78">A32</font></i></a>, <a class="ref" href="matrprob.html#T1" target="_self" title="MATRPROB:th.1">Th1</a></i>;<br/><b>then </b><i><font color="Green" title="E80">A36</font></i>:  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c14">e5</font> = 
<font color="Maroon" title="c6">n1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="rvsum_1.html#T110" title="RVSUM_1:th.110">RVSUM_1:110</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><b>thus </b> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c13">e4</font> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">e4</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E6:72_4"><i><font color="Green" title="E76">A29</font></i></a>, <a class="ref" href="rfinseq.html#T20" title="RFINSEQ:th.20">RFINSEQ:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">e4</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n1</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
<b>by </b><i><a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="matrprob.html#E5:72_4_1_1_2"><i><font color="Green" title="E80">A36</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E10:72_4"/>
 <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c11">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>


</div><b>end;</b></div>
<a NAME="E25:72"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c11">M1</font> holds <br/><font color="Maroon" title="c11">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c14">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:72_5"/><i><font color="Green" title="E73">A37</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c11">M1</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c15">p1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:72_5"/><i><font color="Green" title="E74">A38</font></i>: 
( <font color="Maroon" title="c15">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">M1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c11">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">j</font> )
 <b>by </b><i><a class="txt" href="matrprob.html#E1:72_5"><i><font color="Green" title="E73">A37</font></i></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E4:72_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">M1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c11">M1</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:72_5"><i><font color="Green" title="E73">A37</font></i></a>, <a class="ref" href="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E5:72_5"/><b>then </b><i><font color="Green" title="E75">A39</font></i>: 
( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">M1</font> &amp; <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c11">M1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:72_5"/><i><font color="Green" title="E76">A40</font></i>: 
( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font> or  not <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font> )
 
;<br/>
<a NAME="E7:72_5"/><i><font color="Green" title="E77">A41</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">M1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E8:72_5"/><b>then </b><i><font color="Green" title="E78">A42</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E5:72_5"><i><font color="Green" title="E75">A39</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:72_5"/><i><font color="Green" title="E79">A43</font></i>: 
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E5:72_5"><i><font color="Green" title="E75">A39</font></i></a>, <a class="txt" href="matrprob.html#E7:72_5"><i><font color="Green" title="E77">A41</font></i></a>, <a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:72_5_1"/>
( <font color="Maroon" title="c15">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e2</font> or <font color="Maroon" title="c15">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e1</font> )
 </span><b>by </b><i><a class="txt" href="matrprob.html#E12:72"><i><font color="Green" title="E63">A5</font></i></a>, <a class="txt" href="matrprob.html#E3:72_5"><i><font color="Green" title="E74">A38</font></i></a>, <a class="txt" href="matrprob.html#E6:72_5"><i><font color="Green" title="E76">A40</font></i></a>, <a class="txt" href="matrprob.html#E8:72_5"><i><font color="Green" title="E78">A42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_5_1_1"><b>suppose </b></a><a NAME="E1:72_5_1_1"/>
<font color="Maroon" title="c15">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:72_5_1_1"/>
<font color="Maroon" title="c11">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="matrprob.html#E3:72_5"><i><font color="Green" title="E74">A38</font></i></a>, <a class="txt" href="matrprob.html#E9:72_5"><i><font color="Green" title="E79">A43</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_5_1_2"><b>suppose </b></a><a NAME="E1:72_5_1_2"/><i><font color="Green" title="E80">A44</font></i>: 
<font color="Maroon" title="c15">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e1</font>
 ;<br/><div class="add"><a NAME="E2:72_5_1_2"/>
( <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">m1</font> or  not <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">m1</font> )
 ;<br/><b>hence </b><a NAME="E3:72_5_1_2"/>
<font color="Maroon" title="c11">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="matrprob.html#E8:72"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="matrprob.html#E3:72_5"><i><font color="Green" title="E74">A38</font></i></a>, <a class="txt" href="matrprob.html#E9:72_5"><i><font color="Green" title="E79">A43</font></i></a>, <a class="txt" href="matrprob.html#E1:72_5_1_2"><i><font color="Green" title="E80">A44</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E26:72"/><b>then </b>
<font color="Maroon" title="c11">M1</font> is <a href="matrprob.html#V2" title="MATRPROB:attr.2">m-nonnegative</a>
 
<b>by </b><i><a class="ref" href="matrprob.html#D6" target="_self" title="MATRPROB:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E27:72"/>
 ex <font color="Olive" title="b1">M</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">M</font> is <a href="matrprob.html#V2" title="MATRPROB:attr.2">m-nonnegative</a> &amp;  <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="matrprob.html#E24:72"><i><font color="Green" title="E72">A25</font></i></a></i>;<br/>


</div>
