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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c3">M1</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>;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c3">M1</font> is <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>suppose </b></a><a NAME="E1:20_1_1"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:20_1_1"/><i><font color="Green" title="E10">A3</font></i>: 
(  <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="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/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  = 
<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="c3">M1</font></span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> 
<b>by </b><i><a class="txt" href="matrix_8.html#E1:20"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="matrix_6.html#D5" title="MATRIX_6:def.5">MATRIX_6:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E1:20_1_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="matrix_8.html#E2:20_1_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="matrix_3.html#T24" title="MATRIX_3:th.24">MATRIX_3:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <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_8.html#E1:20"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="matrix_6.html#D5" title="MATRIX_6:def.5">MATRIX_6:def 5</a></i>
;<br/><b>hence </b><a NAME="E4:20_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">M1</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" title="MATRIX_6:def.5">MATRIX_6:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2"><b>suppose </b></a><a NAME="E1:20_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:20_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <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="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></i>;<br/><b>hence </b><a NAME="E3:20_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">M1</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" title="MATRIX_6:def.5">MATRIX_6:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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