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

<span class="kw">set </span><font color="Maroon" title="c3">TR</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">cTR</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">M</font> =  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:71_1_1_1"/>
(  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> or (  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span> &amp;  not <font color="Maroon" title="c2">f</font> is  <a href="matrtop3.html#V3" title="MATRTOP3:attr.3">base_rotation</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop3.html#T37" target="_self" title="MATRTOP3:th.37">Th36</a>, <a class="ref" href="matrtop3.html#T38" target="_self" title="MATRTOP3:th.38">Th37</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:71_1_1_1_1"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a> </span><br/><div class="add"><a NAME="E2:71_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> is  <a href="matrtop3.html#V3" title="MATRTOP3:attr.3">base_rotation</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop3.html#T37" target="_self" title="MATRTOP3:th.37">Th36</a></span>;<br/><span class="kw">hence </span><a NAME="E3:71_1_1_1_1"/>
 <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</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="matrtop3.html#E56"><span class="lab"><font color="Green" title="E56">L</font></span></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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:71_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
(  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span> &amp;  not <font color="Maroon" title="c2">f</font> is  <a href="matrtop3.html#V3" title="MATRTOP3:attr.3">base_rotation</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</font> is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a> </span><br/><div class="add"><a NAME="E2:71_1_1_1_2"/><span class="lab"><font color="Green" title="E58">E</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:71_1_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:71_1_1_1_2_1"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:71_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
;<br/>
<a NAME="E4:71_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A5</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T105" title="JORDAN2C:th.105">JORDAN2C:105</a>, <a class="ref" href="euclid.html#T22" title="EUCLID:th.22">EUCLID:22</a></span>;<br/>
<a NAME="E5:71_1_1_1_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E6:71_1_1_1_2_1"/><span class="kw">then </span>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E7:71_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1_1_2_1"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="txt" href="#E4:71_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="ref" href="funct_1.html#T7" title="FUNCT_1:th.7">FUNCT_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:71_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1_1_2"><span class="lab"><font color="Green" title="E57">A2</font></span></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><span class="kw">end;</span></div><a NAME="E3:71_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A6</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">A</font> =  <a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>);<br/><span class="kw">set </span><font color="Maroon" title="c7">MA</font> =  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p1">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>;<br/><a NAME="E4:71_1_1_1_2"/>
 <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop3.html#D6" target="_self" title="MATRTOP3:def.6">Def6</a></span>;<br/><a NAME="E5:71_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A10</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> is  <a href="matrtop3.html#V3" title="MATRTOP3:attr.3">base_rotation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1_1_2"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E3:71_1_1_1_2"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="ref" href="matrtop3.html#T39" target="_self" title="MATRTOP3:th.39">Th38</a></span>;<br/><a NAME="E6:71_1_1_1_2"/>
(  <a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>) is  <a href="matrix_1.html#V2" title="MATRIX_1:attr.2">diagonal</a>  &amp; <span class="p1">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</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>  <a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_1_1_1_2"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="ref" href="matrtop3.html#T7" target="_self" title="MATRTOP3:th.7">Th15</a></span>;<br/><a NAME="E7:71_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop3.html#T2" target="_self" title="MATRTOP3:th.2">LLL</a></span>;<br/><a NAME="E8:71_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">E0</font></span>: 
(  <a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>) is  <a href="matrix_6.html#V4" title="MATRIX_6:attr.4">Orthogonal</a>  &amp;  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p3">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> 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="#E5:71_1_1_1_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="txt" href="matrtop3.html#E56"><span class="lab"><font color="Green" title="E56">L</font></span></a>, <a class="ref" href="matrix_6.html#D7" title="MATRIX_6:def.7">MATRIX_6:def 7</a></span>;<br/><span class="lab"><font color="Green" title="E62">E2</font></span>:  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p3">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p3">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop3.html#T29" target="_self" title="MATRTOP3:th.29">TA</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop3.html#D6" target="_self" title="MATRTOP3:def.6">Def6</a></span>
;<br/><a NAME="E10:71_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="matrix_6.html#R2" title="MATRIX_6:pred.2">is_reverse_of</a>  <a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_6.html#D4" title="MATRIX_6:def.4">MATRIX_6:def 4</a></span>;<br/><a NAME="E11:71_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">E1</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <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"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_6.html#D2" title="MATRIX_6:def.2">MATRIX_6:def 2</a></span>;<br/><a NAME="E12:71_1_1_1_2"/>
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <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"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</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"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</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"><a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</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/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <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"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <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"><a href="matrtop3.html#K1" title="MATRTOP3:func.1">AxialSymmetry</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#T33" title="MATRIX_3:th.33">MATRIX_3:33</a></span>
<br/>.= 
 <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:71_1_1_1_2"><span class="lab"><font color="Green" title="E63">E1</font></span></a>, <a class="ref" href="matrix_3.html#T18" title="MATRIX_3:th.18">MATRIX_3:18</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:71_1_1_1_2"/>
 <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c2">f</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="#E2:71_1_1_1_2"><span class="lab"><font color="Green" title="E58">E</font></span></a>, <a class="txt" href="#E8:71_1_1_1_2"><span class="lab"><font color="Green" title="E61">E0</font></span></a>, <a class="ref" href="matrix_6.html#T59" title="MATRIX_6:th.59">MATRIX_6:59</a>, <a class="txt" href="#E9:71_1_1_1_2"><span class="lab"><font color="Green" title="E62">E2</font></span></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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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