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<div class="add">

<b>thus </b><a NAME="E1:11"/><i><font color="Green">E19</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<a NAME="E1:11_1"/>
 <a href="sin_cos.html#K33">PI</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,4</span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D30">SIN_COS:def 30</a></i>;<br/>
<a NAME="E2:11_1"/><b>then </b>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="sin_cos.html#K33">PI</a>  &amp;  <a href="sin_cos.html#K33">PI</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E3:11_1"/><b>then </b>
0 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E4:11_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:11"/><i><font color="Green">E20</font></i>: 
<a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="sin_cos.html#K33">PI</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<a NAME="E1:11_2"/>
0 <a href="real_1.html#K3">+</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="xxreal_0.html#NR3" target="_self">&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="real_1.html#K3">+</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>
 
<b>by </b><i><a class="txt" href="comptrig.html#E1:11"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E2:11_2"/>
<a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/>


</div><b>end;</b></div>

<b>hence </b><a NAME="E3:11"/><i><font color="Green">E21</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="sin_cos.html#K33">PI</a> 
 <b>by </b><i><a class="txt" href="comptrig.html#E1:11"><i><font color="Green">E19</font></i></a></i>;<br/>

<a NAME="E4:11"/><b>then </b>
0 <a href="real_1.html#K5">-</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="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K5">-</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>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E5:11"/>
 <a href="real_1.html#K1">-</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="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2
 ;<br/>

<a NAME="E6:11"/><i><font color="Green">E22</font></i>: 
0 <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a> 
 
<b>by </b><i><a class="txt" href="comptrig.html#E3:11"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E7:11"/>
 <a href="sin_cos.html#K33">PI</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 
 ;<br/>

<a NAME="E8:11"/>
0 <a href="real_1.html#K3">+</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="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K3">+</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>
 
<b>by </b><i><a class="txt" href="comptrig.html#E3:11"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E9:11"/>
<a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<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#K33">PI</a> 
 ;<br/>

<a NAME="E10:11"/>
0 <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 0
 
;<br/>
<b>hence </b><a NAME="E11:11"/>
 <a href="real_1.html#K1">-</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="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="comptrig.html#E1:11"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>

<b>thus </b><a NAME="E12:11"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 
 <b>by </b><i><a class="txt" href="comptrig.html#E1:11"><i><font color="Green">E19</font></i></a>, <a class="txt" href="comptrig.html#E2:11"><i><font color="Green">E20</font></i></a>, <a class="txt" href="comptrig.html#E6:11"><i><font color="Green">E22</font></i></a></i>;<br/>

<a NAME="E13:11"/><i><font color="Green">E23</font></i>: 
<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="real_1.html#K3">+</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="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K3">+</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>
 
<b>by </b><i><a class="txt" href="comptrig.html#E2:11"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E14:11"/>
 <a href="sin_cos.html#K33">PI</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<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#K33">PI</a> 
 ;<br/>

<b>thus </b><a NAME="E15:11"/>
<span class="p1">(<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#K33">PI</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 
 <b>by </b><i><a class="txt" href="comptrig.html#E13:11"><i><font color="Green">E23</font></i></a>, <a class="ref" href="comptrig.html#T20" target="_self">Th20</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>

<b>thus </b><a NAME="E16:11"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<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#K33">PI</a> 
 <b>by </b><i><a class="txt" href="comptrig.html#E1:11"><i><font color="Green">E19</font></i></a>, <a class="txt" href="comptrig.html#E2:11"><i><font color="Green">E20</font></i></a>, <a class="txt" href="comptrig.html#E13:11"><i><font color="Green">E23</font></i></a></i>;<br/>


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