<?xml version="1.0"?>
<div class="add">

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

<a NAME="E1:30"/><i><font color="Green">E25</font></i>: 
<a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 0</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 2
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i><a class="ref" href="sin_cos.html#T56">SIN_COS:56</a>, <a class="ref" href="sin_cos.html#D27">SIN_COS:def 27</a></i>
;<br/>
<b>hence </b><a NAME="E2:30_1"/>
<a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:30"/>
<a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<a NAME="E1:30_2"/>
( <a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <a href="sin_cos.html#K27">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">p</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="sin_cos.html#T59">SIN_COS:59</a></i>;<br/>
<a NAME="E2:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">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">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E3:30_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>hence </b><a NAME="E4:30_2"/>
<a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30"/>
( <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i/>;<br/>


</div>
