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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><a NAME="E1:14"/>
( <a href="sin_cos.html#K35" title="SIN_COS:func.35">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>) <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#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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>) )
 ;<br/>

<a NAME="E2:14"/><b>then </b>
( <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">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>) <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#R1" title="XXREAL_0:pred.1">&lt;</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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E3:14"/><b>then </b>
<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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> ,<span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E4:14"/><b>then </b>
<a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="comptrig.html#T25" title="COMPTRIG:th.25">COMPTRIG:25</a></i>;<br/>
<a NAME="E5:14"/><b>then </b>
<a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T8" target="_self" title="SIN_COS6:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E6:14"/>
 <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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="sin_cos.html#D21" title="SIN_COS:def.21">SIN_COS:def 21</a></i>;<br/>


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