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<b>let </b><font color="Maroon" title="c1">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 implies  <a href="power.html#K5" title="POWER:func.5">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos7.html#K4" title="SIN_COS7:func.4">tanh"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:87"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="power.html#K5" title="POWER:func.5">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos7.html#K4" title="SIN_COS7:func.4">tanh"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E2:87"/><b>then </b><i><font color="Green" title="E57">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<b>then </b>2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos7.html#K4" title="SIN_COS7:func.4">tanh"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
 <a href="power.html#K5" title="POWER:func.5">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T114" title="XCMPLX_1:th.114">XCMPLX_1:114</a></i>
<br/>.= 
 <a href="power.html#K5" title="POWER:func.5">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K6" title="BINOP_2:func.6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:87"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T128" title="XCMPLX_1:th.128">XCMPLX_1:128</a></i>
<br/>.= 
 <a href="power.html#K5" title="POWER:func.5">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="#E2:87"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T55" title="XCMPLX_1:th.55">XCMPLX_1:55</a></i>
<br/>.= 
 <a href="power.html#K5" title="POWER:func.5">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">x</font>

;<br/>
<b>hence </b><a NAME="E4:87"/>
 <a href="power.html#K5" title="POWER:func.5">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos7.html#K4" title="SIN_COS7:func.4">tanh"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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