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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> , <font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:11"/><i><font color="Green">E23</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 
 <b>and </b><br/><a NAME="E3:11"/><i><font color="Green">E29</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:11"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 
 <b>and </b><br/><a NAME="E5:11"/><i><font color="Green">E31</font></i>: 
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K20">sin</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E6:11"/><i><font color="Green">E32</font></i>: 
 <a href="sin_cos.html#K23">cos</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K23">cos</a> <font color="Maroon">s</font>
 ;<br/>

<i><font color="Green">E33</font></i>:  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i>, , <a class="ref" href="sin_cos.html#T32">SIN_COS:32</a></i>
;<br/>
<i><font color="Green">E48</font></i>:  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i>, , <a class="ref" href="sin_cos.html#T32">SIN_COS:32</a></i>
;<br/>
<i><font color="Green">E49</font></i>:  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
0
<b>by </b><i>, </i>
;<br/>
<i><font color="Green">E50</font></i>:  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
0
<b>by </b><i>, </i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_5"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> or <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> or <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 </span><b>by </b><i><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="11_5_1"><b>suppose </b></a><a NAME="E1:11_5_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:11_5_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 0
 ;<br/><a NAME="E3:11_5_1"/><b>then </b><i><font color="Green">E51</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E4:11_5_1"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 0 <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">s</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E5:11_5_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:11_5_1"/><b>then </b>
( <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  )
 <b>by </b><i>, <a class="ref" href="complex2.html#T2" target="_self">Th2</a>, <a class="ref" href="comptrig.html#T33">COMPTRIG:33</a></i>;<br/><b>hence </b><a NAME="E7:11_5_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="sin_cos.html#T82">SIN_COS:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_5_2"><b>suppose </b></a><a NAME="E1:11_5_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:11_5_2"/><b>then </b>
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 0
 ;<br/><a NAME="E3:11_5_2"/><b>then </b><i><font color="Green">E52</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E4:11_5_2"/>
<font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 0 <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">r</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E5:11_5_2"/><b>then </b>
<font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:11_5_2"/><b>then </b>
( <font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  )
 <b>by </b><i>, <a class="ref" href="complex2.html#T2" target="_self">Th2</a>, <a class="ref" href="comptrig.html#T33">COMPTRIG:33</a></i>;<br/><b>hence </b><a NAME="E7:11_5_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="sin_cos.html#T82">SIN_COS:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_5_3"><b>suppose </b></a><a NAME="E1:11_5_3"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_5_3"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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