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

<b>let </b><font color="Maroon" title="c1">n</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"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">M1</font>, <font color="Olive" title="b3">M2</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font>  st <font color="Maroon" title="c1">n</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>  &amp; <font color="Olive" title="b2">M1</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> &amp; <font color="Olive" title="b3">M2</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> holds <br/>( <font color="Olive" title="b2">M1</font> <a href="matrix_6.html#R1" title="MATRIX_6:pred.1">commutes_with</a> <font color="Olive" title="b3">M2</font> iff <font color="Olive" title="b2">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b3">M2</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> )</span><br/><b>let </b><font color="Maroon" title="c2">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">M1</font>, <font color="Olive" title="b2">M2</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st <font color="Maroon" title="c1">n</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>  &amp; <font color="Olive" title="b1">M1</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> &amp; <font color="Olive" title="b2">M2</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> holds <br/>( <font color="Olive" title="b1">M1</font> <a href="matrix_6.html#R1" title="MATRIX_6:pred.1">commutes_with</a> <font color="Olive" title="b2">M2</font> iff <font color="Olive" title="b1">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">M2</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> )</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#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">n</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>  &amp; <font color="Maroon" title="c3">M1</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> &amp; <font color="Maroon" title="c4">M2</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> implies ( <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#R1" title="MATRIX_6:pred.1">commutes_with</a> <font color="Maroon" title="c4">M2</font> iff <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> ) )</span><br/>







<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c1">n</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>  &amp; <font color="Maroon" title="c3">M1</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> &amp; <font color="Maroon" title="c4">M2</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#R1" title="MATRIX_6:pred.1">commutes_with</a> <font color="Maroon" title="c4">M2</font> iff <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> )</span><br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
( <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M1</font> &amp; <font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M2</font> )
 
<b>by </b><i><a class="ref" href="matrix_6.html#D5" target="_self" title="MATRIX_6:def.5">Def5</a></i>;<br/>
<a NAME="E3:31"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c3">M1</font> is   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c4">M2</font> is   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</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> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <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> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<b>thus </b><a NAME="E4:31"/>
( <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#R1" title="MATRIX_6:pred.1">commutes_with</a> <font color="Maroon" title="c4">M2</font> implies <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> implies <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#R1" title="MATRIX_6:pred.1">commutes_with</a> <font color="Maroon" title="c4">M2</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#R1" title="MATRIX_6:pred.1">commutes_with</a> <font color="Maroon" title="c4">M2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  = 
<font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">M1</font>
<b>by </b><i><a class="txt" href="matrix_6.html#E1:31"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="matrix_6.html#E2:31"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="matrix_6.html#E3:31"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="matrix_3.html#T24" title="MATRIX_3:th.24">MATRIX_3:24</a></i>
<br/>.= 
<font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font>
<b>by </b><i><a class="txt" href="matrix_6.html#E1:31_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="matrix_6.html#D1" target="_self" title="MATRIX_6:def.1">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E3:31_1"/>
<font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a>
 <b>by </b><i><a class="ref" href="matrix_6.html#D5" target="_self" title="MATRIX_6:def.5">Def5</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><b>end;</b></div>

<b>assume </b><a NAME="E5:31"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#R1" title="MATRIX_6:pred.1">commutes_with</a> <font color="Maroon" title="c4">M2</font></span><br/>

<font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">M1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> 
<b>by </b><i><a class="txt" href="matrix_6.html#E1:31"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="matrix_6.html#E2:31"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="matrix_6.html#E3:31"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="matrix_3.html#T24" title="MATRIX_3:th.24">MATRIX_3:24</a></i>
<br/>.= 
<font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font>
<b>by </b><i><a class="txt" href="matrix_6.html#E5:31"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="matrix_6.html#D5" target="_self" title="MATRIX_6:def.5">Def5</a></i>
;<br/>
<b>hence </b><a NAME="E7:31"/>
<font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#R1" title="MATRIX_6:pred.1">commutes_with</a> <font color="Maroon" title="c4">M2</font>
 <b>by </b><i><a class="ref" href="matrix_6.html#D1" target="_self" title="MATRIX_6:def.1">Def1</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>
