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<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:99"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/>

<a NAME="E2:99"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E3:99"/><b>then </b><i><font color="Green">E81</font></i>: 
<a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T57" target="_self">Th57</a></i>;<br/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E100"><i><font color="Green">Lemma77</font></i></a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="sin_cos.html#E102"><i><font color="Green">Lemma78</font></i></a></i>
;<br/>
<a NAME="E5:99"/><b>then </b><i><font color="Green">E82</font></i>: 
<a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T74">XCMPLX_1:74</a></i>;<br/>
<b>hence </b><a NAME="E6:99"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E3:99"><i><font color="Green">E81</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>

<b>thus </b><a NAME="E7:99"/>
<a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="sin_cos.html#E3:99"><i><font color="Green">E81</font></i></a>, <a class="txt" href="sin_cos.html#E5:99"><i><font color="Green">E82</font></i></a>, <a class="ref" href="real_2.html#T164">REAL_2:164</a></i>;<br/>


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