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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">p</font>, <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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>  st <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b2">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> holds <br/> ex <font color="Olive" title="b3">f</font> being   <a href="topreal9.html#V1" title="TOPREAL9:attr.1">homogeneous</a>   <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="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> st <br/>( <font color="Olive" title="b3">f</font> is  <a href="matrtop3.html#V2" title="MATRTOP3:attr.2">rotation</a>  &amp; <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> &amp; (  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Olive" title="b3">f</font> <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>) or  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Olive" title="b3">f</font> <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><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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 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="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> implies  ex <font color="Olive" title="b1">f</font> being   <a href="topreal9.html#V1" title="TOPREAL9:attr.1">homogeneous</a>   <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="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> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="matrtop3.html#V2" title="MATRTOP3:attr.2">rotation</a>  &amp; <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp; (  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Olive" title="b1">f</font> <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>) or  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Olive" title="b1">f</font> <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><br/>





<span class="kw">set </span><font color="Maroon" title="c4">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">assume </span><span class="kw">that </span><br/><a NAME="E1:73"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:73"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="topreal9.html#V1" title="TOPREAL9:attr.1">homogeneous</a>   <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="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> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="matrtop3.html#V2" title="MATRTOP3:attr.2">rotation</a>  &amp; <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp; (  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Olive" title="b1">f</font> <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>) or  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Olive" title="b1">f</font> <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><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_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="E1:73_1"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> or <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1"><span class="kw">suppose </span></a><a NAME="E1:73_1_1"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="topreal9.html#V1" title="TOPREAL9:attr.1">homogeneous</a>   <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="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> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="matrtop3.html#V2" title="MATRTOP3:attr.2">rotation</a>  &amp; <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp; (  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Olive" title="b1">f</font> <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>) or  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Olive" title="b1">f</font> <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><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c5">I</font> =  <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>; <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="c5">I</font> is  <a href="matrtop3.html#V2" title="MATRTOP3:attr.2">rotation</a>  &amp; <font color="Maroon" title="c5">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp; (  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c5">I</font> <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>) or  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c5">I</font> <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><br/><a NAME="E2:73_1_1"/>
 <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> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p1">(<span class="default"><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>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="matrtop1.html#T33" title="MATRTOP1:th.33">MATRTOP1:33</a></span>;<br/><span class="kw">hence </span><a NAME="E3:73_1_1"/>
( <font color="Maroon" title="c5">I</font> is  <a href="matrtop3.html#V2" title="MATRTOP3:attr.2">rotation</a>  &amp; <font color="Maroon" title="c5">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp; (  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c5">I</font> <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>) or  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c5">I</font> <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="txt" href="#E1:73"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E1:73_1_1"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="ref" href="matrtop3.html#D6" target="_self" title="MATRTOP3:def.6">Def6</a>, <a class="ref" href="funct_1.html#T17" title="FUNCT_1:th.17">FUNCT_1:17</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="73_1_2"><span class="kw">suppose </span></a><a NAME="E1:73_1_2"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="topreal9.html#V1" title="TOPREAL9:attr.1">homogeneous</a>   <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="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> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="matrtop3.html#V2" title="MATRTOP3:attr.2">rotation</a>  &amp; <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp; (  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Olive" title="b1">f</font> <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>) or  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Olive" title="b1">f</font> <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><br/><div class="add"><a NAME="E2:73_1_2"/><span class="lab"><font color="Green" title="E60">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E3:73_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A6</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><a NAME="E4:73_1_2"/><span class="lab"><font color="Green" title="E62">A7</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">f</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> (1,1)</span>)</span> as   <a href="topreal9.html#V1" title="TOPREAL9:attr.1">homogeneous</a>   <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>   <a href="matrtop3.html#V2" title="MATRTOP3:attr.2">rotation</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="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="#E1:73"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="matrtop3.html#T27" target="_self" title="MATRTOP3:th.27">Th25</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">f</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="c5">f</font> is  <a href="matrtop3.html#V2" title="MATRTOP3:attr.2">rotation</a>  &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp; (  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c5">f</font> <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>) or  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c5">f</font> <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><br/><a NAME="E6:73_1_2"/><span class="lab"><font color="Green" title="E63">A8</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" title="XREAL_1:th.63">XREAL_1:63</a></span>;<br/><span class="lab"><font color="Green" title="E64">A9</font></span>: <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_2"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T55" title="RVSUM_1:th.55">RVSUM_1:55</a></span>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T73" title="RVSUM_1:th.73">RVSUM_1:73</a></span>
;<br/><a NAME="E8:73_1_2"/><span class="lab"><font color="Green" title="E65">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E9:73_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A11</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73_1_2"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T55" title="RVSUM_1:th.55">RVSUM_1:55</a></span>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T73" title="RVSUM_1:th.73">RVSUM_1:73</a></span>
;<br/><a NAME="E11:73_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E6:73_1_2"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="txt" href="#E7:73_1_2"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="ref" href="square_1.html#T28" title="SQUARE_1:th.28">SQUARE_1:28</a></span>;<br/><a NAME="E12:73_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> = 
<span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
<br/>.= 
<span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E4:73_1_2"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="ref" href="matrtop3.html#T9" target="_self" title="MATRTOP3:th.9">Th17</a></span>
<br/>.= 
<font color="Maroon" title="c3">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_2"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E3:73_1_2"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="txt" href="#E9:73_1_2"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E11:73_1_2"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="ref" href="square_1.html#T40" title="SQUARE_1:th.40">SQUARE_1:40</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:73_1_2"/>
( <font color="Maroon" title="c5">f</font> is  <a href="matrtop3.html#V2" title="MATRTOP3:attr.2">rotation</a>  &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp; (  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c5">f</font> <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>) or  <a href="matrtop3.html#K3" title="MATRTOP3:func.3">AutMt</a> <font color="Maroon" title="c5">f</font> <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="txt" href="#E1:73"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="matrtop3.html#D6" target="_self" title="MATRTOP3:def.6">Def6</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>

</div>
