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<span class="kw">let </span><font color="Maroon" title="c1">x</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"> (  <a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies (  <a href="sin_cos4.html#K3" title="SIN_COS4:func.3">cosec</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos4.html#K4" title="SIN_COS4:func.4">sec</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K3" title="SIN_COS4:func.3">cosec</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="sin_cos4.html#K3" title="SIN_COS4:func.3">cosec</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K2" title="SIN_COS4:func.2">cot</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:14"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 <a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:14"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
 <a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="sin_cos4.html#K3" title="SIN_COS4:func.3">cosec</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos4.html#K4" title="SIN_COS4:func.4">sec</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K3" title="SIN_COS4:func.3">cosec</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="sin_cos4.html#K3" title="SIN_COS4:func.3">cosec</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K2" title="SIN_COS4:func.2">cot</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )</span><br/>

<span class="lab"><font color="Green" title="E11">A3</font></span>:  <a href="sin_cos4.html#K3" title="SIN_COS4:func.3">cosec</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="sin_cos5.html#T6" target="_self" title="SIN_COS5:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T57" title="XCMPLX_1:th.57">XCMPLX_1:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T62" title="XCMPLX_1:th.62">XCMPLX_1:62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos4.html#K2" title="SIN_COS4:func.2">cot</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T57" title="XCMPLX_1:th.57">XCMPLX_1:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos4.html#K2" title="SIN_COS4:func.2">cot</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="xcmplx_1.html#T50" title="XCMPLX_1:th.50">XCMPLX_1:50</a>, <a class="ref" href="xcmplx_1.html#T89" title="XCMPLX_1:th.89">XCMPLX_1:89</a></span>
;<br/>
 <a href="sin_cos4.html#K3" title="SIN_COS4:func.3">cosec</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="sin_cos5.html#T6" target="_self" title="SIN_COS5:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T57" title="XCMPLX_1:th.57">XCMPLX_1:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos4.html#K4" title="SIN_COS4:func.4">sec</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="sin_cos5.html#T11" target="_self" title="SIN_COS5:th.11">Th11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos4.html#K4" title="SIN_COS4:func.4">sec</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K4" title="SIN_COS4:func.4">sec</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos4.html#K4" title="SIN_COS4:func.4">sec</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos4.html#K4" title="SIN_COS4:func.4">sec</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T74" title="XCMPLX_1:th.74">XCMPLX_1:74</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos4.html#K4" title="SIN_COS4:func.4">sec</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K4" title="SIN_COS4:func.4">sec</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T77" title="XCMPLX_1:th.77">XCMPLX_1:77</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos4.html#K4" title="SIN_COS4:func.4">sec</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K3" title="SIN_COS4:func.3">cosec</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T106" title="XCMPLX_1:th.106">XCMPLX_1:106</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:14"/>
(  <a href="sin_cos4.html#K3" title="SIN_COS4:func.3">cosec</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos4.html#K4" title="SIN_COS4:func.4">sec</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K3" title="SIN_COS4:func.3">cosec</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="sin_cos4.html#K3" title="SIN_COS4:func.3">cosec</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos4.html#K1" title="SIN_COS4:func.1">tan</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K2" title="SIN_COS4:func.2">cot</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">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/>


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