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

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">M2</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>;<br/>







<b>assume </b><a NAME="E1:45"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:45"/><i><font color="Green">E37</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E3:45"/><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E4:45"/><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E5:45"/><i><font color="Green">E40</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E6:45"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E7:45"/><i><font color="Green">E42</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<b>thus </b><a NAME="E8:45"/>
( <font color="Maroon">M1</font> <a href="matrix_6.html#R1" target="_self">commutes_with</a> <font color="Maroon">M2</font> implies <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">M1</font> <a href="matrix_6.html#R1" target="_self">commutes_with</a> <font color="Maroon">M2</font>
 ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="matrix_4.html#T62">MATRIX_4:62</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_4.html#T63">MATRIX_4:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_4.html#T63">MATRIX_4:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="matrix_3.html#T5">MATRIX_3:5</a></i>
;<br/>
<b>hence </b><a NAME="E3:45_1"/>
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E9:45"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span>
 ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="matrix_4.html#T62">MATRIX_4:62</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2" target="_self">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_4.html#T63">MATRIX_4:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_4.html#T63">MATRIX_4:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="matrix_3.html#T5">MATRIX_3:5</a></i>
;<br/>
<a NAME="E11:45"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="matrix_3.html#T5">MATRIX_3:5</a></i>;<br/>
<a NAME="E12:45"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="matrix_3.html#T5">MATRIX_3:5</a></i>;<br/>
<a NAME="E13:45"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="matrix_4.html#T4">MATRIX_4:4</a></i>;<br/>
<a NAME="E14:45"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="matrix_3.html#T4">MATRIX_3:4</a></i>;<br/>
<a NAME="E15:45"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrix_3.html#T4">MATRIX_3:4</a></i>;<br/>
<a NAME="E16:45"/><b>then </b>
<font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font>
 
<b>by </b><i>, , <a class="ref" href="matrix_4.html#T4">MATRIX_4:4</a></i>;<br/>
<b>hence </b><a NAME="E17:45"/>
<font color="Maroon">M1</font> <a href="matrix_6.html#R1" target="_self">commutes_with</a> <font color="Maroon">M2</font>
 <b>by </b><i/>;<br/>


</div>
