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<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/>





<a NAME="E1:53"/><i><font color="Green">E35</font></i>: 
(  <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">M1</font> <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#K5" target="_self">~</a> </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#K5" target="_self">~</a> </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="E2:53"/><i><font color="Green">E37</font></i>: 
(  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </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_1.html#K5">@</a> </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="E3:53"/>
( <font color="Maroon">M1</font> is <a href="matrix_6.html#V1" target="_self">invertible</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font> implies <font color="Maroon">M1</font> is <a href="matrix_6.html#V4" target="_self">Orthogonal</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">M1</font> is <a href="matrix_6.html#V1" target="_self">invertible</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:53_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a>  <a href="matrix_6.html#R2" target="_self">is_reverse_of</a> <font color="Maroon">M1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:53_1"/><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#K5" target="_self">~</a> </span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</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#K5" target="_self">~</a> </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_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</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#K5" target="_self">~</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:53_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a> </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#K4" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a> </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_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</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#K5" target="_self">~</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrix_3.html#T35">MATRIX_3:35</a></i>;<br/>
<a NAME="E5:53_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a> </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#K4" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </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#K4" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="matrix_3.html#T35">MATRIX_3:35</a></i>;<br/>
<a NAME="E6:53_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a> </span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </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#K4" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a> </span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:53_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a> </span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:53_1"/><b>then </b>
<font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#T21">MATRIX_3:21</a></i>;<br/>
<a NAME="E9:53_1"/><b>then </b>
<font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> 
 
<b>by </b><i><a class="ref" href="matrix_3.html#T21">MATRIX_3:21</a></i>;<br/>
<b>hence </b><a NAME="E10:53_1"/>
<font color="Maroon">M1</font> is <a href="matrix_6.html#V4" target="_self">Orthogonal</a>
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E4:53"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">M1</font> is <a href="matrix_6.html#V4" target="_self">Orthogonal</a>
 ;<br/>

<a NAME="E5:53"/><b>then </b>
( <font color="Maroon">M1</font> is <a href="matrix_6.html#V1" target="_self">invertible</a> &amp; <font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E6:53"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a> </span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">M1</font> <a href="matrix_6.html#K5" target="_self">~</a>  <a href="matrix_6.html#R2" target="_self">is_reverse_of</a> <font color="Maroon">M1</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:53"/>
( <font color="Maroon">M1</font> is <a href="matrix_6.html#V1" target="_self">invertible</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4" target="_self">*</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font> )
 <b>by </b><i>, , </i>;<br/>


</div>
