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<b>let </b><font color="Maroon">r</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">i</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="E1:24"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>)
 ;<br/>

<b>assume </b><a NAME="E3:24"/><i><font color="Green">E24</font></i>: 
 <a href="sin_cos.html#K23">cos</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E4:24"/><b>then </b><i><font color="Green">E31</font></i>: 
( ( <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>) <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> or <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35">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">i</font>) ) &amp; ( <span class="p1">(<span class="default"><a href="sin_cos.html#K35">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">i</font>) <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> or <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>) ) )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E11"><i><font color="Green">Lemma46</font></i></a>, <a class="txt" href="sin_cos6.html#E12"><i><font color="Green">Lemma47</font></i></a></i>;<br/>
<a NAME="E5:24"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>) <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>)
 
<b>by </b><i><a class="ref" href="comptrig.html#T21">COMPTRIG:21</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:24"/><b>then </b>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35">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">i</font>) or <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>) or <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>) )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E16"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="sin_cos6.html#E19"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:24"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35">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">i</font>) or <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>) )
 <b>by </b><i><a class="txt" href="sin_cos6.html#E17"><i><font color="Green">Lemma52</font></i></a>, <a class="txt" href="sin_cos6.html#E18"><i><font color="Green">Lemma53</font></i></a>, <a class="txt" href="sin_cos6.html#E13"><i><font color="Green">Lemma48</font></i></a></i>;<br/>


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