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<span class="kw">let </span><font color="Maroon" title="c1">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c1">r</font> holds <br/> <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<span class="p2"><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">&lt;*</a><span class="default"><span class="p3">(<span class="default">(<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b3">b</font></span>)</span>,<font color="Olive" title="b2">a</font>)</span>)</span>,<span class="p3">(<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>,<font color="Olive" title="b1">n</font>)</span>)</span></span><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">*&gt;</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span>)</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></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c1">r</font> holds <br/> <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<span class="p2"><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">&lt;*</a><span class="default"><span class="p3">(<span class="default">(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b2">b</font></span>)</span>,<font color="Olive" title="b1">a</font>)</span>)</span>,<span class="p3">(<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>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">*&gt;</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span>)</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></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c1">r</font> implies  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<span class="p2"><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">&lt;*</a><span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c3">a</font>)</span>)</span>,<span class="p3">(<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>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">*&gt;</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span>)</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> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c5">A</font> = (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c3">a</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">ONE</font> =  <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="c2">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">B</font> =  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<span class="p1"><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">&lt;*</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c3">a</font>)</span>)</span>,<span class="p2">(<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>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span>);<br/>
<a NAME="E1:23"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
( <font color="Maroon" title="c2">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>  or <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<a NAME="E2:23"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E3:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A2</font></span>: 
(  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</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>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">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="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="c2">n</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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="matrixr2.html#T41" title="MATRIXR2:th.41">MATRIXR2:41</a>, <a class="ref" href="matrix_7.html#T16" title="MATRIX_7:th.16">MATRIX_7:16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:23"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c1">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<span class="p2"><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">&lt;*</a><span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c3">a</font>)</span>)</span>,<span class="p3">(<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>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">*&gt;</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span>)</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></span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E18">A3</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>

<br/>.= 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c3">a</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T13" title="MATRIX_9:th.13">MATRIX_9:13</a></span>
;<br/>
<a NAME="E6:23"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
(  <a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r</font> &amp;  <a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D17" title="SIN_COS:def.17">SIN_COS:def 17</a>, <a class="ref" href="sin_cos.html#D19" title="SIN_COS:def.19">SIN_COS:def 19</a></span>;<br/>
<span class="kw">thus </span> <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<span class="p2"><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">&lt;*</a><span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c3">a</font>)</span>)</span>,<span class="p3">(<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>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">*&gt;</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c3">a</font>)</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<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>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#T52" title="MATRIXJ1:th.52">MATRIXJ1:52</a></span>
<br/>.= 
 <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="sin_cos.html#T28" title="SIN_COS:th.28">SIN_COS:28</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/>


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