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<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="E1:31"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">M1</font> is <a href="matrix_6.html#V2" target="_self">symmetric</a> &amp; <font color="Maroon">M2</font> is <a href="matrix_6.html#V2" target="_self">symmetric</a> )
 ;<br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">M2</font> <a href="matrix_1.html#K5">@</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:31"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">M1</font> is   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font> &amp; <font color="Maroon">M2</font> is   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</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> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</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/>
<b>thus </b><a NAME="E4:31"/>
( <font color="Maroon">M1</font> <a href="matrix_6.html#R1" target="_self">commutes_with</a> <font color="Maroon">M2</font> implies <font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font> is <a href="matrix_6.html#V2" target="_self">symmetric</a> )
 <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">E39</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#K4" target="_self">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K5">@</a>  = 
<font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font>
<b>by </b><i>, , , <a class="ref" href="matrix_3.html#T24">MATRIX_3:24</a></i>
<br/>.= 
<font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E3:31_1"/>
<font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font> is <a href="matrix_6.html#V2" target="_self">symmetric</a>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E5:31"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font> is <a href="matrix_6.html#V2" target="_self">symmetric</a>
 ;<br/>

<font color="Maroon">M2</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font> = 
<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_1.html#K5">@</a> 
<b>by </b><i>, , , <a class="ref" href="matrix_3.html#T24">MATRIX_3:24</a></i>
<br/>.= 
<font color="Maroon">M1</font> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M2</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E7:31"/>
<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>
