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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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</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_cos.html#K24" title="SIN_COS:func.24">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">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> &amp;  <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</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">2 <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_cos.html#K24" title="SIN_COS:func.24">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="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp;  <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</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> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</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"><span class="p3">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">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><br/>

<span class="lab"><font color="Green" title="E6">A1</font></span>:  <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</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="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">x</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">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#T80" title="SIN_COS:th.80">SIN_COS:80</a></span>
;<br/>
<span class="kw">then </span> <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</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> = 
<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_cos.html#K24" title="SIN_COS:func.24">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">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="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1

<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_cos.html#K24" title="SIN_COS:func.24">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">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>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T32" title="SIN_COS:th.32">SIN_COS:32</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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"><span class="p4">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">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>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E3:7"/>
(  <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</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_cos.html#K24" title="SIN_COS:func.24">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">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> &amp;  <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</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">2 <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_cos.html#K24" title="SIN_COS:func.24">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="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp;  <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</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> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</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"><span class="p3">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">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="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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