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<div class="add">

<b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_2.html#D3">RFUNCT_2:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K19">sin</a> </span>)</span>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K19">sin</a> </span>)</span>
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font>
 ;<br/>

<a NAME="E4:31"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K19">sin</a>  &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K19">sin</a>  )
 
<b>by </b><i><a class="txt" href="comptrig.html#E2:31"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:31"/><b>then </b><i><font color="Green">E37</font></i>: 
( <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&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#K35">PI</a>  &amp; <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&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#K35">PI</a>  )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E6:31"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">r1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T30">SIN_COS:30</a></i>;<br/>
<a NAME="E7:31"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>;<br/>
<a NAME="E8:31"/><b>then </b><i><font color="Green">E38</font></i>: 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<b>set </b><font color="Maroon">r3</font> = <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E9:31"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T30">SIN_COS:30</a></i>;<br/>
<a NAME="E10:31"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>;<br/>
<a NAME="E11:31"/><b>then </b><i><font color="Green">E39</font></i>: 
( <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<a NAME="E12:31"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T30">SIN_COS:30</a></i>;<br/>
<a NAME="E13:31"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>;<br/>
<a NAME="E14:31"/><b>then </b><i><font color="Green">E40</font></i>: 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<a NAME="E15:31"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T228">XREAL_1:228</a></i>;<br/>
<a NAME="E16:31"/><b>then </b>
( <a href="sin_cos.html#K35">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"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <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#K35">PI</a>  )
 
<b>by </b><i><a class="txt" href="comptrig.html#E3:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:31"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K19">sin</a>  )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a>, <a class="ref" href="sin_cos.html#T27">SIN_COS:27</a></i>;<br/>
<a NAME="E18:31"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K19">sin</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1"/>
( <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> or <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> )
 </span><b>by </b><i><a class="txt" href="comptrig.html#E3:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_1_1"/><i><font color="Green">E45</font></i>: 
<a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 ;<br/><div class="add"><a NAME="E2:31_1_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_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="E1:31_1_1_1_1_1"/>
( <font color="Maroon">r2</font> <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#K35">PI</a>  or <font color="Maroon">r2</font> <a href="hidden.html#R1">=</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#K35">PI</a>  )
 </span><b>by </b><i><a class="txt" href="comptrig.html#E3:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_1_1_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">r2</font> <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#K35">PI</a> 
 ;<br/><div class="add"><a NAME="E2:31_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">r1</font> <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#K35">PI</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:31_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="txt" href="comptrig.html#E14:31"><i><font color="Green">E40</font></i></a>, , <a class="txt" href="comptrig.html#E15:31"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E4:31_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K19">sin</a> </span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K19">sin</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:31_1_1_1_1_1_1"/>
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font>
 <b>by </b><i>, , <a class="ref" href="rfunct_2.html#D3">RFUNCT_2:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_1_1_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">r2</font> <a href="hidden.html#R1">=</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#K35">PI</a> 
 ;<br/><div class="add"><a NAME="E2:31_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="txt" href="comptrig.html#E14:31"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E3:31_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K19">sin</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:31_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <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#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="txt" href="comptrig.html#E8:31"><i><font color="Green">E38</font></i></a>, <a class="txt" href="comptrig.html#E11:31"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="rfunct_2.html#D3">RFUNCT_2:def 3</a></i>;<br/><b>assume </b><a NAME="E5:31_1_1_1_1_1_2"/>
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font>
 ;<br/><b>hence </b><a NAME="E6:31_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="comptrig.html#E4:31"><i><font color="Green">E36</font></i></a>, <a class="txt" href="comptrig.html#E15:31"><i><font color="Green">E42</font></i></a>, <a class="txt" href="comptrig.html#E18:31"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="sin_cos.html#T81">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="E4:31_1_1_1"/>
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_1_2"/><i><font color="Green">E86</font></i>: 
<a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_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="E1:31_1_1_2_1_1"/>
( <font color="Maroon">r2</font> <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#K35">PI</a>  or <font color="Maroon">r2</font> <a href="hidden.html#R1">=</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#K35">PI</a>  )
 </span><b>by </b><i><a class="txt" href="comptrig.html#E3:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_1_2_1_1_1"/>
<font color="Maroon">r2</font> <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#K35">PI</a> 
 ;<br/><div class="add"><a NAME="E2:31_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="txt" href="comptrig.html#E14:31"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E3:31_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><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 class="p2">(<span class="default"><span class="p3">(<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></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K19">sin</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:31_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="comptrig.html#E8:31"><i><font color="Green">E38</font></i></a>, <a class="txt" href="comptrig.html#E11:31"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="rfunct_2.html#D3">RFUNCT_2:def 3</a></i>;<br/><b>assume </b><a NAME="E5:31_1_1_2_1_1_1"/>
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font>
 ;<br/><b>hence </b><a NAME="E6:31_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="comptrig.html#E4:31"><i><font color="Green">E36</font></i></a>, <a class="txt" href="comptrig.html#E5:31"><i><font color="Green">E37</font></i></a>, <a class="txt" href="comptrig.html#E14:31"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="sin_cos.html#T81">SIN_COS:81</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_1_2_1_1_2"/>
<font color="Maroon">r2</font> <a href="hidden.html#R1">=</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#K35">PI</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_1_2_1_1_2"/>
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="txt" href="comptrig.html#E14:31"><i><font color="Green">E40</font></i></a>, <a class="ref" href="sin_cos.html#T81">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="E3:31_1_1_2"/>
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E20:31"/>
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font>
 ;<br/>


</div>
