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

<span class="kw">let </span><font color="Maroon" title="c1">p</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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 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">q</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> <br/> for <font color="Olive" title="b2">M</font> being  <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>  holds <br/> ( <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Olive" title="b1">q</font></span>)</span> iff (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">M</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">p</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b2">M</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="Olive" title="b1">q</font> &amp; ( for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">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="b3">i</font>,<font color="Olive" title="b4">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="Olive" title="b2">M</font> holds <br/><font color="Olive" title="b2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">j</font></span>)</span> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">q</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 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">M</font> being  <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>  holds <br/> ( <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font></span>)</span> iff (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">M</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">p</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">M</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="c2">q</font> &amp; ( for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">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="b2">i</font>,<font color="Olive" title="b3">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="Olive" title="b1">M</font> holds <br/><font color="Olive" title="b1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">j</font></span>)</span> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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"> ( <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font></span>)</span> iff (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</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">p</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</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="c2">q</font> &amp; ( 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="c3">M</font> holds <br/><font color="Maroon" title="c3">M</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> ) ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">M1</font> =  <a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">M2</font> =  <a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font>;<br/>
<a NAME="E1:27"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">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="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#D9" title="MATRIXR1:def.9">MATRIXR1:def 9</a></span>;<br/>
<a NAME="E2:27"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></span>;<br/>
<a NAME="E3:27"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</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="E4:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#D9" title="MATRIXR1:def.9">MATRIXR1:def 9</a></span>;<br/>
<a NAME="E5:27"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</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">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></span>;<br/>
<div><span class="kw">hereby </span> <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="c3">M</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">p</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</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="c2">q</font> &amp; ( 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="c3">M</font> holds <br/><font color="Maroon" title="c3">M</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> ) implies <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:27_1"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font></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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</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">p</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</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="c2">q</font> &amp; ( 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="c3">M</font> holds <br/><font color="Maroon" title="c3">M</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> ) )</span><br/><a NAME="E2:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</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="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="matrprob.html#T39" title="MATRPROB:th.39">MATRPROB:39</a></span>;<br/><span class="kw">thus </span><a NAME="E3:27_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</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">p</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</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="c2">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="matrprob.html#T39" title="MATRPROB:th.39">MATRPROB:39</a></span>; <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">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="c3">M</font> holds <br/><font color="Maroon" title="c3">M</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span></span><br/><a NAME="E4:27_1"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="matrprob.html#T39" title="MATRPROB:th.39">MATRPROB:39</a></span>;<br/><span class="kw">thus </span><a NAME="E5:27_1"/>
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="c3">M</font> holds <br/><font color="Maroon" title="c3">M</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">j</font></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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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="c3">M</font> implies <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:27_1_1"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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="c3">M</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="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span></span><br/>

<a NAME="E2:27_1_1"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="matrprob.html#T12" title="MATRPROB:th.12">MATRPROB:12</a></span>;<br/>
<a NAME="E3:27_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A11</font></span>: 
<font color="Maroon" title="c6">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="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E28">A7</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="E4:27_1_1"/>
<font color="Maroon" title="c7">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="c3">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="matrprob.html#T12" title="MATRPROB:th.12">MATRPROB:12</a></span>;<br/>
<a NAME="E5:27_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<font color="Maroon" title="c7">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="c2">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E4:27_1"><span class="lab"><font color="Green" title="E29">A8</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="E6:27_1_1"/>
<font color="Maroon" title="c6">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="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E2:27_1_1"><span class="lab"><font color="Green" title="E31">A10</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">then </span><span class="lab"><font color="Green" title="E34">A13</font></span>:  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c6">i</font> = 
<span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c6">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>
<br/>.= 
<span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E3:27_1_1"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="ref" href="matrixr1.html#D9" title="MATRIXR1:def.9">MATRIXR1:def 9</a></span>
;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font></span>)</span>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E1:27_1_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="matrprob.html#T39" title="MATRPROB:th.39">MATRPROB:39</a></span>
<br/>.= 
<span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1_1"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E7:27_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="entropy1.html#T20" target="_self" title="ENTROPY1:th.20">Th20</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <span class="p2"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span>,<span class="p2"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#D17" title="RVSUM_1:def.17">RVSUM_1:def 17</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T89" title="RVSUM_1:th.89">RVSUM_1:89</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T103" title="RVSUM_1:th.103">RVSUM_1:103</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>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:27"/><span class="lab"><font color="Green" title="E27">A14</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</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">p</font>
 <span class="kw">and </span><br/><a NAME="E8:27"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</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="c2">q</font>
 <span class="kw">and </span><br/><a NAME="E9:27"/><span class="lab"><font color="Green" title="E29">A16</font></span>: 
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="c3">M</font> holds <br/><font color="Maroon" title="c3">M</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span>
 ; <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="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font></span>)</span></span><br/>

<a NAME="E10:27"/>
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="c3">M</font> holds <br/><font color="Maroon" title="c3">M</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Olive" title="b1">i</font></span>)</span> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font></span>)</span>,<font color="Olive" title="b2">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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="c3">M</font> implies <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font></span>)</span>,<font color="Maroon" title="c7">j</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:27_2"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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="c3">M</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="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font></span>)</span>,<font color="Maroon" title="c7">j</font></span>)</span></span><br/>

<a NAME="E2:27_2"/>
<font color="Maroon" title="c7">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="c3">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_2"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="matrprob.html#T12" title="MATRPROB:th.12">MATRPROB:12</a></span>;<br/>
<a NAME="E3:27_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<font color="Maroon" title="c7">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="c2">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E28">A15</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="E4:27_2"/><span class="lab"><font color="Green" title="E32">A19</font></span>: 
<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_2"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="matrprob.html#T12" title="MATRPROB:th.12">MATRPROB:12</a></span>;<br/>
<a NAME="E5:27_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A20</font></span>: 
<font color="Maroon" title="c6">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="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E27">A14</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="E6:27_2"/>
<font color="Maroon" title="c6">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="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E4:27_2"><span class="lab"><font color="Green" title="E32">A19</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">then </span><span class="lab"><font color="Green" title="E34">A21</font></span>: <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c6">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="matrixr1.html#D9" title="MATRIXR1:def.9">MATRIXR1:def 9</a></span>
<br/>.= 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c6">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_2"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>
;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E1:27_2"><span class="lab"><font color="Green" title="E30">A17</font></span></a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T103" title="RVSUM_1:th.103">RVSUM_1:103</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <span class="p2"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span>,<span class="p2"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T89" title="RVSUM_1:th.89">RVSUM_1:89</a></span>
<br/>.= 
<span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#D17" title="RVSUM_1:def.17">RVSUM_1:def 17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font></span>)</span>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_2"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="txt" href="#E7:27_2"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="entropy1.html#T20" target="_self" title="ENTROPY1:th.20">Th20</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>
<span class="kw">hence </span><a NAME="E11:27"/>
<font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="matrprob.html#T39" title="MATRPROB:th.39">MATRPROB:39</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>
