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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#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 <span class="p1">(<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> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:13"/>
 <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"> <span class="p1">(<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> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span><br/>

<a NAME="E2:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<span class="p1">(<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="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  <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">by</span> <span class="lab"><a class="ref" href="square_1.html#T12" title="SQUARE_1:th.12">SQUARE_1:12</a></span>;<br/>
<span class="p1">(<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> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  = 
<span class="p1">(<span class="default">1 <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"><span class="p2">(<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="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T76" title="XCMPLX_1:th.76">XCMPLX_1:76</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</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">x</font></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T29" title="SIN_COS:th.29">SIN_COS:29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><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> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><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> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span></span>)</span>
<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/>.= 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><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> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
<br/>.= 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</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> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T76" title="XCMPLX_1:th.76">XCMPLX_1:76</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:13"/>
<span class="p1">(<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> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="square_1.html#K5" title="SQUARE_1:func.5">^2</a></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/>


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