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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</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>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Olive" title="b1">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font> holds <br/><font color="Olive" title="b1">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) holds <br/> <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be    <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>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) implies  <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">and </span><br/><a NAME="E2:47"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> 1
 <span class="kw">and </span><br/><a NAME="E3:47"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_1_1"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:47_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E41">A1</font></span></a></span>;<br/><span class="kw">hence </span> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a></span>
<br/>.= 
 <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">M</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="matrprob.html#T23" title="MATRPROB:th.23">MATRPROB:23</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_1_2"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:47_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c3">q</font> =  <a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <font color="Maroon" title="c1">M</font>;<br/><a NAME="E3:47_1_2"/><span class="lab"><font color="Green" title="E46">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrprob.html#D1" title="MATRPROB:def.1">MATRPROB:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">qq</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> <span class="kw"> such that </span><br/><a NAME="E5:47_1_2"/><span class="lab"><font color="Green" title="E47">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:47_1_2"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
<font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <span class="kw">and </span><br/><a NAME="E7:47_1_2"/><span class="lab"><font color="Green" title="E49">A10</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <font color="Maroon" title="c1">M</font></span>)</span> holds <br/><font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:47_1_2"/><span class="lab"><font color="Green" title="E50">A11</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_2"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="entropy1.html#T9" target="_self" title="ENTROPY1:th.9">Th9</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><span class="lab"><font color="Green" title="E51">A12</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">qq</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_2"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="ref" href="matrprob.html#D1" title="MATRPROB:def.1">MATRPROB:def 1</a></span>
<br/>.= 
 <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="c2">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="matrprob.html#D1" title="MATRPROB:def.1">MATRPROB:def 1</a></span>
;<br/><a NAME="E10:47_1_2"/><span class="lab"><font color="Green" title="E52">A13</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">qq</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E11:47_1_2"/><span class="lab"><font color="Green" title="E53">A14</font></span>: 
 <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="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrprob.html#D1" title="MATRPROB:def.1">MATRPROB:def 1</a></span>;<br/><a NAME="E12:47_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A15</font></span>: 
 <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="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><a NAME="E13:47_1_2"/>
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">qq</font> holds <br/><font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">qq</font> implies <font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 );<br/>
<a NAME="E1:47_1_2_2"/><span class="lab"><font color="Green" title="E55">A16</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_2_2_1"/><span class="lab"><font color="Green" title="E56">A17</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:47_1_2_2_1"/><span class="lab"><font color="Green" title="E57">A18</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">qq</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E3:47_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A19</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">qq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47_1_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E4:47_1_2_2_1"/><span class="lab"><font color="Green" title="E59">A20</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E3:47_1_2"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E11:47_1_2"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="txt" href="#E5:47_1_2"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E10:47_1_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="txt" href="#E2:47_1_2_2_1"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:47_1_2_2_1"/><span class="lab"><font color="Green" title="E60">A21</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_2"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E5:47_1_2"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E10:47_1_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="txt" href="#E2:47_1_2_2_1"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:47_1_2_2_1"/>
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47_1_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="txt" href="#E2:47_1_2_2_1"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E7:47_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A22</font></span>: 
( <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_1_2_2_1_1"/>
( <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_1_2_2_1"><span class="lab"><font color="Green" title="E61">A22</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E62">A23</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:47_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E63">A24</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_2"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:47_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A25</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:47_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E65">A26</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:47_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E63">A24</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">M</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_1_2"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E1:47_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E62">A23</font></span></a>, <a class="txt" href="#E2:47_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E63">A24</font></span></a>, <a class="ref" href="matrprob.html#T20" title="MATRPROB:th.20">MATRPROB:20</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E64">A25</font></span></a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E12:47_1_2"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="txt" href="#E4:47_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E65">A26</font></span></a>, <a class="ref" href="matrprob.html#D1" title="MATRPROB:def.1">MATRPROB:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:47_1_2_2_1_1_1"/>
<font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E62">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E62">A27</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:47_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E63">A28</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">qq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_2_2_1"><span class="lab"><font color="Green" title="E58">A19</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:47_1_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A29</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_2"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="ref" href="matrprob.html#D1" title="MATRPROB:def.1">MATRPROB:def 1</a></span>;<br/><a NAME="E4:47_1_2_2_1_1_2"/>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">qq</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E62">A27</font></span></a>, <a class="txt" href="#E2:47_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E63">A28</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E5:47_1_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A30</font></span>: 
<font color="Maroon" title="c5">k</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="c2">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:47_1_2"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_2"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E7:47_1_2"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="txt" href="#E1:47_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E62">A27</font></span></a>, <a class="txt" href="#E2:47_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E63">A28</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47_1_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="txt" href="#E1:47_1_2_2_1"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E1:47_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E62">A27</font></span></a>, <a class="txt" href="#E2:47_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E63">A28</font></span></a>, <a class="txt" href="#E5:47_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E65">A30</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="matrprob.html#D1" title="MATRPROB:def.1">MATRPROB:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_2"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E5:47_1_2"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E10:47_1_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="txt" href="#E2:47_1_2_2_1"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="ref" href="matrprob.html#T20" title="MATRPROB:th.20">MATRPROB:20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_2_2_1"><span class="lab"><font color="Green" title="E60">A21</font></span></a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:47_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E62">A27</font></span></a>, <a class="txt" href="#E3:47_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E64">A29</font></span></a>, <a class="ref" href="entropy1.html#T37" target="_self" title="ENTROPY1:th.37">Th37</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_2_2_1"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="ref" href="matrprob.html#D1" title="MATRPROB:def.1">MATRPROB:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:47_1_2_2_1_1_2"/>
<font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:47_1_2_2"/><span class="lab"><font color="Green" title="E56">A31</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47_1_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E3:47_1_2_2"/>
for <font color="Olive" title="b1">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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:47_1_2_2"><span class="lab"><font color="Green" title="E56">A31</font></span></a>, <a class="txt" href="#E1:47_1_2_2"><span class="lab"><font color="Green" title="E55">A16</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:47_1_2_2"/>
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">qq</font> holds <br/><font color="Maroon" title="c4">qq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E14:47_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A32</font></span>: 
<font color="Maroon" title="c4">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:47_1_2"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/><a NAME="E15:47_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_2"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/><a NAME="E16:47_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A33</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</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="c2">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E11:47_1_2"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">thus </span> <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">M</font> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <font color="Maroon" title="c1">M</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrprob.html#D3" title="MATRPROB:def.3">MATRPROB:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47_1_2"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="txt" href="#E14:47_1_2"><span class="lab"><font color="Green" title="E55">A32</font></span></a>, <a class="ref" href="matrprob.html#D1" title="MATRPROB:def.1">MATRPROB:def 1</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:47_1_2"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="ref" href="matrprob.html#D1" title="MATRPROB:def.1">MATRPROB:def 1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
