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<b>let </b><font color="Maroon" title="c1">r1</font>, <font color="Maroon" title="c2">r2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_2.html#D2" title="RFUNCT_2:def.2">RFUNCT_2:def 2</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c1">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</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#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span>
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c2">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</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#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span>
 <b>and </b><br/><a NAME="E3:33"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r2</font>
 ;<br/>

<a NAME="E4:33"/><i><font color="Green" title="E34">A4</font></i>: 
( <font color="Maroon" title="c1">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</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#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c1">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  &amp; <font color="Maroon" title="c2">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</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#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c2">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  )
 
<b>by </b><i><a class="txt" href="comptrig.html#E1:33"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="comptrig.html#E2:33"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:33"/><b>then </b><i><font color="Green" title="E35">A5</font></i>: 
(  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r1</font> &amp; <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <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>  &amp;  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> &amp; <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <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>  )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">r3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2;<br/>
<a NAME="E6:33"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T30" title="SIN_COS:th.30">SIN_COS:30</a></i>;<br/>
<a NAME="E7:33"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#D23" title="SIN_COS:def.23">SIN_COS:def 23</a></i>;<br/>
<a NAME="E8:33"/><b>then </b><i><font color="Green" title="E36">A6</font></i>: 
( <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <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 &amp; <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>
<a NAME="E9:33"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T30" title="SIN_COS:th.30">SIN_COS:30</a></i>;<br/>
<a NAME="E10:33"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#D23" title="SIN_COS:def.23">SIN_COS:def 23</a></i>;<br/>
<a NAME="E11:33"/><b>then </b><i><font color="Green" title="E37">A7</font></i>: 
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">r2</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="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>
<a NAME="E12:33"/><i><font color="Green" title="E38">A8</font></i>: 
( <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r2</font> )
 
<b>by </b><i><a class="txt" href="comptrig.html#E3:33"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="xreal_1.html#T228" title="XREAL_1:th.228">XREAL_1:228</a></i>;<br/>
<a NAME="E13:33"/><b>then </b>
(  <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"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <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>  )
 
<b>by </b><i><a class="txt" href="comptrig.html#E5:33"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:33"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <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#K4" title="REAL_1:func.4">*</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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a>, <a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</a></i>;<br/>
<a NAME="E15:33"/><b>then </b><i><font color="Green" title="E39">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <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#K4" title="REAL_1:func.4">*</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> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1"/>
(  <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> <font color="Maroon" title="c1">r1</font> or  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r1</font> )
 </span><b>by </b><i><a class="txt" href="comptrig.html#E5:33"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1"/><i><font color="Green" title="E40">A10</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> <font color="Maroon" title="c1">r1</font>
 ;<br/><div class="add"><a NAME="E2:33_1_1_1"/><b>then </b><i><font color="Green" title="E41">A11</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> <font color="Maroon" title="c2">r2</font>
 <b>by </b><i><a class="txt" href="comptrig.html#E3:33"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_1"/>
( <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <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>  or <font color="Maroon" title="c2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <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><b>by </b><i><a class="txt" href="comptrig.html#E5:33"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_1_1"/><i><font color="Green" title="E42">A12</font></i>: 
<font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <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> 
 ;<br/><div class="add"><a NAME="E2:33_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <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> 
 <b>by </b><i><a class="txt" href="comptrig.html#E3:33"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:33_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c1">r1</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#K4" title="REAL_1:func.4">*</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> &amp; <font color="Maroon" title="c2">r2</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#K4" title="REAL_1:func.4">*</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="txt" href="comptrig.html#E1:33_1_1_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="comptrig.html#E2:33_1_1_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="comptrig.html#E1:33_1_1_1_1_1_1"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E4:33_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c1">r1</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#K4" title="REAL_1:func.4">*</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> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span> &amp; <font color="Maroon" title="c2">r2</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#K4" title="REAL_1:func.4">*</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> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="comptrig.html#E4:33"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_1_1_1_1"/>
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r1</font>
 <b>by </b><i><a class="txt" href="comptrig.html#E3:33"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="comptrig.html#T38" target="_self" title="COMPTRIG:th.38">Th38</a>, <a class="ref" href="rfunct_2.html#D2" title="RFUNCT_2:def.2">RFUNCT_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_1_2"/><i><font color="Green" title="E42">A13</font></i>: 
<font color="Maroon" title="c2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <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> 
 ;<br/><div class="add"><a NAME="E2:33_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">r1</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#K4" title="REAL_1:func.4">*</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="txt" href="comptrig.html#E3:33"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="comptrig.html#E1:33_1_1_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E3:33_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">r1</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#K4" title="REAL_1:func.4">*</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> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span>
 <b>by </b><i><a class="txt" href="comptrig.html#E4:33"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:33_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E43">A14</font></i>: 
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r1</font>
 <b>by </b><i><a class="txt" href="comptrig.html#E12:33"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="comptrig.html#E15:33"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="comptrig.html#T38" target="_self" title="COMPTRIG:th.38">Th38</a>, <a class="ref" href="rfunct_2.html#D2" title="RFUNCT_2:def.2">RFUNCT_2:def 2</a></i>;<br/><b>assume </b><a NAME="E5:33_1_1_1_1_1_2"/>
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r1</font>
 ;<br/><b>hence </b><a NAME="E6:33_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="comptrig.html#E8:33"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="comptrig.html#E1:33_1_1_1_1_1_2"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="comptrig.html#E4:33_1_1_1_1_1_2"><i><font color="Green" title="E43">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="sin_cos.html#T81" title="SIN_COS:th.81">SIN_COS:81</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_1"/>
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_2"/><i><font color="Green" title="E40">A15</font></i>: 
 <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_2_1_1"/>
( <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <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>  or <font color="Maroon" title="c2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <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><b>by </b><i><a class="txt" href="comptrig.html#E5:33"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_1_1"/>
<font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <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> 
 ;<br/><div class="add"><a NAME="E2:33_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">r2</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#K4" title="REAL_1:func.4">*</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="txt" href="comptrig.html#E3:33"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="comptrig.html#E1:33_1_1_2"><i><font color="Green" title="E40">A15</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E3:33_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">r2</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#K4" title="REAL_1:func.4">*</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> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span>
 <b>by </b><i><a class="txt" href="comptrig.html#E4:33"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:33_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E41">A16</font></i>: 
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="comptrig.html#E12:33"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="comptrig.html#E15:33"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="comptrig.html#T38" target="_self" title="COMPTRIG:th.38">Th38</a>, <a class="ref" href="rfunct_2.html#D2" title="RFUNCT_2:def.2">RFUNCT_2:def 2</a></i>;<br/><b>assume </b><a NAME="E5:33_1_1_2_1_1_1"/>
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r1</font>
 ;<br/><b>hence </b><a NAME="E6:33_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="comptrig.html#E8:33"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="comptrig.html#E11:33"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="comptrig.html#E1:33_1_1_2"><i><font color="Green" title="E40">A15</font></i></a>, <a class="txt" href="comptrig.html#E4:33_1_1_2_1_1_1"><i><font color="Green" title="E41">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="ref" href="sin_cos.html#T81" title="SIN_COS:th.81">SIN_COS:81</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_1_2"/>
<font color="Maroon" title="c2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <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> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_2_1_1_2"/>
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r1</font>
 <b>by </b><i><a class="txt" href="comptrig.html#E1:33_1_1_2"><i><font color="Green" title="E40">A15</font></i></a>, <a class="ref" href="sin_cos.html#T81" title="SIN_COS:th.81">SIN_COS:81</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_2"/>
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r1</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:33"/>
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r1</font>
 ;<br/>


</div>
