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

<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">M1</font>, <font color="Olive" title="b3">M2</font> being  <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">M1</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="b3">M2</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b3">M2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">M1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Olive" title="b3">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Olive" title="b2">M1</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Olive" title="b3">M2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M1</font>, <font color="Olive" title="b2">M2</font> being  <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">M1</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="b2">M2</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b2">M2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">M1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Olive" title="b2">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Olive" title="b1">M1</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Olive" title="b2">M2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">M1</font>, <font color="Maroon" title="c4">M2</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M1</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="c4">M2</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M2</font></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E5">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M1</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="c4">M2</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M2</font></span>)</span></span><br/>

<a NAME="E2:18"/><i><font color="Green" title="E6">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span></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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <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"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E3:18"/><i><font color="Green" title="E7">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</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="c3">M1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M1</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D3" title="MATRIX_3:def.3">MATRIX_3:def 3</a></i>;<br/>
<a NAME="E4:18"/><i><font color="Green" title="E8">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</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="c3">M1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M1</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E5:18"/><i><font color="Green" title="E9">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M2</font></span>)</span></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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <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"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D3" title="MATRIX_3:def.3">MATRIX_3:def 3</a></i>;<br/>
<a NAME="E6:18"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> )</span><br/>

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E10">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><br/>

<a NAME="E2:18_1"/><i><font color="Green" title="E11">A7</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:18"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="matrix_4.html#T55" title="MATRIX_4:th.55">MATRIX_4:55</a></i>;<br/>
<a NAME="E3:18_1"/><i><font color="Green" title="E12">A8</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M1</font>
 
<b>by </b><i><a class="txt" href="#E3:18"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="matrix_4.html#T55" title="MATRIX_4:th.55">MATRIX_4:55</a></i>;<br/>
<a NAME="E4:18_1"/><i><font color="Green" title="E13">A9</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M1</font>
 
<b>by </b><i><a class="txt" href="#E4:18"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="matrix_4.html#T55" title="MATRIX_4:th.55">MATRIX_4:55</a></i>;<br/>
<a NAME="E5:18_1"/><i><font color="Green" title="E14">A10</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M1</font>
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="matrix_4.html#T55" title="MATRIX_4:th.55">MATRIX_4:55</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="#E2:18_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="#E3:18_1"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="#E4:18_1"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="matrix_3.html#D3" title="MATRIX_3:def.3">MATRIX_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="#E2:18_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="#E3:18_1"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="#E2:18_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="#E3:18_1"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="#E5:18_1"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></i>
;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> = 
<font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="#E2:18_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="#E3:18_1"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="matrix_3.html#D3" title="MATRIX_3:def.3">MATRIX_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>
<b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="#E2:18_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E8:18_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:18"/>
<font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:18"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="#E3:18"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="#E4:18"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="#E5:18"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
