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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="sin_cos5.html#K2" title="SIN_COS5:func.2">sech</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos5.html#K2" title="SIN_COS5:func.2">sech</a> <font color="Maroon" title="c1">x</font> &amp;  <a href="sin_cos5.html#K2" title="SIN_COS5:func.2">sech</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<a NAME="E1:12"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
2 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">exp_R</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">exp_R</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos8.html#E11"><span class="lab"><font color="Green" title="E11">Lm9</font></span></a>, <a class="ref" href="xreal_1.html#T183" onmouseover="rs('xreal_1/T183')" onmouseout="rh()">XREAL_1:183</a></span>;<br/>
<a NAME="E2:12"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos2.html#K3" title="SIN_COS2:func.3">cosh</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos2.html#T15" onmouseover="rs('sin_cos2/T15')" onmouseout="rh()">SIN_COS2:15</a></span>;<br/>
<a NAME="E3:12"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos2.html#D4" onmouseover="rs('sin_cos2/D4')" onmouseout="rh()">SIN_COS2:def 4</a></span>;<br/>
<a NAME="E4:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a></span>;<br/>
 <a href="sin_cos5.html#K2" title="SIN_COS5:func.2">sech</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos8.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="sin_cos5.html#D2" onmouseover="rs('sin_cos5/D2')" onmouseout="rh()">SIN_COS5:def 2</a></span>
<br/>.= 
1

;<br/>
<span class="kw">hence </span><a NAME="E6:12"/>
(  <a href="sin_cos5.html#K2" title="SIN_COS5:func.2">sech</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos5.html#K2" title="SIN_COS5:func.2">sech</a> <font color="Maroon" title="c1">x</font> &amp;  <a href="sin_cos5.html#K2" title="SIN_COS5:func.2">sech</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="sin_cos5.html#T36" onmouseover="rs('sin_cos5/T36')" onmouseout="rh()">SIN_COS5:36</a>, <a class="ref" href="sin_cos5.html#D2" onmouseover="rs('sin_cos5/D2')" onmouseout="rh()">SIN_COS5:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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