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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="int_1.html#V1">integer</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 <b>and </b><br/><a NAME="E3:39"/><i><font color="Green">E67</font></i>: 
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E4:39"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E1:39"><i><font color="Green">E65</font></i></a>, <a class="txt" href="sin_cos6.html#E2:39"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E5:39"/><i><font color="Green">E69</font></i>: 
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T4" target="_self">Th4</a></i>;<br/>
 <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)</span>)</span> = 
 <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>( <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font>)</span>)</span>

<br/>.= 
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="complex2.html#T9">COMPLEX2:9</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="sin_cos6.html#E3:39"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sin_cos6.html#E5:39"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>
;<br/>
<b>hence </b><a NAME="E7:39"/>
contradiction
 <b>by </b><i><a class="txt" href="sin_cos6.html#E4:39"><i><font color="Green">E68</font></i></a>, <a class="ref" href="sin_cos6.html#T30" target="_self">Th30</a></i>;<br/>


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