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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>, <font color="Olive" title="b4">M3</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#V4" title="MATRIX_6:attr.4">Orthogonal</a>  &amp; <font color="Olive" title="b3">M2</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a>  &amp; <font color="Olive" title="b4">M3</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a>  holds <br/><span class="p1">(<span class="default"><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></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b4">M3</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a> </span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M1</font>, <font color="Olive" title="b2">M2</font>, <font color="Olive" title="b3">M3</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#V4" title="MATRIX_6:attr.4">Orthogonal</a>  &amp; <font color="Olive" title="b2">M2</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a>  &amp; <font color="Olive" title="b3">M3</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a>  holds <br/><span class="p1">(<span class="default"><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></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b3">M3</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">M1</font>, <font color="Maroon" title="c4">M2</font>, <font color="Maroon" title="c5">M3</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#V4" title="MATRIX_6:attr.4">Orthogonal</a>  &amp; <font color="Maroon" title="c4">M2</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a>  &amp; <font color="Maroon" title="c5">M3</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a>  implies <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_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">M3</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a>  )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:68"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:68"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
( <font color="Maroon" title="c3">M1</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a>  &amp; <font color="Maroon" title="c4">M2</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a>  )
 <span class="kw">and </span><br/><a NAME="E3:68"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c5">M3</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">M3</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a> </span><br/>

<a NAME="E4:68"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c5">M3</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="matrix_6.html#D7" target="_self" title="MATRIX_6:def.7">Def7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">M5</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">M3</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span></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_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">M4</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_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">M3</font>;<br/>
<a NAME="E5:68"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
(  <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E6:68"/>
( <font color="Maroon" title="c3">M1</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <font color="Maroon" title="c4">M2</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="matrix_6.html#D7" target="_self" title="MATRIX_6:def.7">Def7</a></span>;<br/>
<a NAME="E7:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A6</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<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_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">M3</font></span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">M3</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span></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_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span> &amp; <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_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">M3</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="matrix_6.html#T56" target="_self" title="MATRIX_6:th.56">Th58</a></span>;<br/>
<a NAME="E8:68"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
(  <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; <font color="Maroon" title="c5">M3</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="c5">M3</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="matrix_6.html#D7" target="_self" title="MATRIX_6:def.7">Def7</a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E9:68"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M3</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E10:68"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
( <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> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  &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> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="matrix_6.html#D7" target="_self" title="MATRIX_6:def.7">Def7</a></span>;<br/>
<a NAME="E11:68"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</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="c4">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E12:68"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E13:68"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M3</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="c5">M3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<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_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">M3</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="c5">M3</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"><span class="p2">(<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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E11:68"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="matrix_3.html#T22" title="MATRIX_3:th.22">MATRIX_3:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">M3</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E5:68"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E10:68"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="matrix_3.html#T22" title="MATRIX_3:th.22">MATRIX_3:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">M3</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span></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_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E12:68"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="matrix_3.html#T33" title="MATRIX_3:th.33">MATRIX_3:33</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:68"/>
<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_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">M3</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="matrix_6.html#D7" target="_self" title="MATRIX_6:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
