<?xml version="1.0"?>
<div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</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> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">i</font></span>)</span> holds <br/> <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span><br/><b>let </b><font color="Maroon" title="c2">i</font> be  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</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> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">i</font></span>)</span> implies  <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:44"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font>
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>)
 <b>and </b><br/><a NAME="E3:44"/><i><font color="Green" title="E56">A3</font></i>: 
 <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:44"/><i><font color="Green" title="E57">A4</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>) &amp; <font color="Maroon" title="c1">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>) )
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E2:44"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E5:44"/><i><font color="Green" title="E58">A5</font></i>: 
 <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T5" target="_self" title="SIN_COS6:th.5">Th5</a></i>;<br/>
 <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>)</span>)</span> = 
 <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon">H<sub>1</sub></font>( <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">i</font>)</span>)</span>

<br/>.= 
 <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <font color="Maroon" title="c1">r</font>
<b>by </b><i><a class="ref" href="complex2.html#T10" title="COMPLEX2:th.10">COMPLEX2:10</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
<b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="#E5:44"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>
;<br/>
<b>hence </b><a NAME="E7:44"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:44"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="sin_cos6.html#T35" target="_self" title="SIN_COS6:th.35">Th35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
