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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<i><font color="Green">E14</font></i>:  <a href="sin_cos5.html#K2">sech</a> 0 = 
1 <a href="real_1.html#K6">/</a> 1
<b>by </b><i><a class="txt" href="sin_cos8.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="ref" href="sin_cos5.html#D2">SIN_COS5:def 2</a></i>
<br/>.= 
1

;<br/>
<a NAME="E2:12"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="sin_cos2.html#T15">SIN_COS2:15</a></i>;<br/>
<a NAME="E3:12"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="sin_cos2.html#D4">SIN_COS2:def 4</a></i>;<br/>
<a NAME="E4:12"/><b>then </b><i><font color="Green">E15</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/>
<a NAME="E5:12"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="txt" href="sin_cos8.html#E12"><i><font color="Green">Lemma12</font></i></a></i>;<br/>
<a NAME="E6:12"/><b>then </b>
2 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T185">XREAL_1:185</a></i>;<br/>
<b>hence </b><a NAME="E7:12"/>
(  <a href="sin_cos5.html#K2">sech</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos5.html#K2">sech</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="sin_cos5.html#K2">sech</a> 0 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="sin_cos5.html#T36">SIN_COS5:36</a>, <a class="txt" href="sin_cos8.html#E1:12"><i><font color="Green">E14</font></i></a>, <a class="txt" href="sin_cos8.html#E4:12"><i><font color="Green">E15</font></i></a>, <a class="ref" href="sin_cos5.html#D2">SIN_COS5:def 2</a></i>;<br/>


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