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<div class="add">

<b>thus </b><a NAME="E1:1"/><i><font color="Green" title="E2">A1</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  &amp;  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  &amp; <span class="p1">(<span class="default">3 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<a NAME="E1:1_1"/>
 <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,4</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D32" title="SIN_COS:def.32">SIN_COS:def 32</a></i>;<br/>
<a NAME="E2:1_1"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  &amp;  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 )
 
<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="E3:1_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E4:1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:1"/><i><font color="Green" title="E3">A2</font></i>: 
<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  &amp; <span class="p1">(<span class="default">3 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<a NAME="E1:1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E2:1_2"/>
<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:1"/><i><font color="Green" title="E4">A3</font></i>: 
 <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">3 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_3"><b>proof </b></a><div class="add">

<a NAME="E1:1_3"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos2.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E2:1_3"/>
 <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<a NAME="E4:1"/><i><font color="Green" title="E5">A4</font></i>: 
<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 
;<br/>
<a NAME="E5:1"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 
;<br/>
<b>hence </b><a NAME="E6:1"/>
<span class="p1">(<span class="default">3 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 <b>by </b><i><a class="txt" href="sin_cos2.html#E3:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="sin_cos2.html#E4:1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
