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<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> ;<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> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green" title="E64">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="E2:40"/><i><font color="Green" title="E65">A2</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>)
 <b>and </b><br/><a NAME="E3:40"/><i><font color="Green" title="E66">A3</font></i>: 
 <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</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
 ;<br/>

<a NAME="E4:40"/><i><font color="Green" title="E67">A4</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>) )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E1:40"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="sin_cos6.html#E2:40"><i><font color="Green" title="E65">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="E5:40"/><i><font color="Green" title="E68">A5</font></i>: 
 <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</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#T3" target="_self" title="SIN_COS6:th.3">Th3</a></i>;<br/>
 <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">i</font>)</span>)</span> = 
 <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K20" title="SIN_COS:func.20">sin</a> <font color="Maroon" title="c1">r</font>
<b>by </b><i><a class="ref" href="complex2.html#T9" title="COMPLEX2:th.9">COMPLEX2:9</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
<b>by </b><i><a class="txt" href="sin_cos6.html#E3:40"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="sin_cos6.html#E5:40"><i><font color="Green" title="E68">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="E7:40"/>
contradiction
 <b>by </b><i><a class="txt" href="sin_cos6.html#E4:40"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="sin_cos6.html#T31" target="_self" title="SIN_COS6:th.31">Th31</a></i>;<br/>


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