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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"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies (  <a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )</span><br/>

<a NAME="E1:5"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 <a href="sin_cos.html#K23" title="SIN_COS:func.23">exp_R</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T55" onmouseover="rs('sin_cos/T55')" onmouseout="rh()">SIN_COS:55</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:5"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<a NAME="E3:5"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<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="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K23" title="SIN_COS:func.23">exp_R</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="sin_cos7.html#T29" onmouseover="rs('sin_cos7/T29')" onmouseout="rh()">SIN_COS7:29</a>, <a class="ref" href="xcmplx_1.html#T199" onmouseover="rs('xcmplx_1/T199')" onmouseout="rh()">XCMPLX_1:199</a></span>;<br/>
<a NAME="E4:5"/><span class="kw">then </span>
<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#K6" title="XCMPLX_0:func.6">-</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> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="taylor_1.html#T4" onmouseover="rs('taylor_1/T4')" onmouseout="rh()">TAYLOR_1:4</a></span>;<br/>
<a NAME="E5:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 <a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos8.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/>
<a NAME="E6:5"/>
 <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<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></span>;<br/>
<a NAME="E7:5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">x</font></span>)</span> <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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="xcmplx_1.html#T50" onmouseover="rs('xcmplx_1/T50')" onmouseout="rh()">XCMPLX_1:50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:5"/>
(  <a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos8.html#T1" target="_self" onmouseover="rs('sin_cos8/T1')" onmouseout="rh()">Th1</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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