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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2" title="COMSEQ_3:func.2">#N</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default">$1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p3">(<span class="default">$1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">z</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:58"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
 for <font color="Olive" title="b1">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2" title="COMSEQ_3:func.2">#N</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">z</font></span>)</span></span>)</span></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">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:58_1"/>
 for <font color="Olive" title="b1">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2" title="COMSEQ_3:func.2">#N</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">z</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2" title="COMSEQ_3:func.2">#N</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">cn</font> = <a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">sn</font> = <a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">c1</font> = <a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">s1</font> = <a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font>;<br/>
<span class="lab"><font color="Green" title="E46">A3</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2" title="COMSEQ_3:func.2">#N</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K1" title="COMSEQ_3:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D2" title="COMSEQ_3:def.2">COMSEQ_3:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p5">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K1" title="COMSEQ_3:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2" title="COMSEQ_3:func.2">#N</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D2" title="COMSEQ_3:def.2">COMSEQ_3:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<span class="p1">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos3.html#T4" target="_self" title="SIN_COS3:th.4">Th4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos3.html#T6" target="_self" title="SIN_COS3:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p5">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p5">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E3:58_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2" title="COMSEQ_3:func.2">#N</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1"><span class="lab"><font color="Green" title="E46">A3</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>
<span class="kw">hence </span><a NAME="E3:58_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <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="E2:58"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos3.html#T21" target="_self" title="SIN_COS3:th.21">Th21</a>, <a class="ref" href="sin_cos3.html#T23" target="_self" title="SIN_COS3:th.23">Th23</a>, <a class="ref" href="complex1.html#D4" title="COMPLEX1:def.4">COMPLEX1:def 4</a>, <a class="ref" href="comseq_3.html#T11" title="COMSEQ_3:th.11">COMSEQ_3:11</a></span>;<br/>
<a NAME="E3:58"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E44">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:58"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b2">z</font></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b2">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2" title="COMSEQ_3:func.2">#N</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K2" title="SIN_COS3:func.2">cos_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">z</font></span>)</span></span>)</span> <a href="sin_cos3.html#K5" title="SIN_COS3:func.5">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">z</font></span>)</span></span>)</span></span>)</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>
