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

<a NAME="E1:32"/>
<a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos2.html#T15">SIN_COS2:15</a></i>;<br/>
<a NAME="E2:32"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos2.html#D4">SIN_COS2:def 4</a></i>;<br/>
<a NAME="E3:32"/>
 <a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E4:32"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T15">XREAL_1:15</a></i>;<br/>
<a NAME="E5:32"/>
<span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E6:32"/>
( <span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>


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