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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="sin_cos.html#K30" title="SIN_COS:func.30">exp_R</a> 1</span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><a href="sin_cos.html#K30" title="SIN_COS:func.30">exp_R</a> 1</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c1">x</font> &amp; <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c1">x</font> &amp; <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c1">x</font> )</span><br/>

<span class="kw">thus </span><span class="lab"><font color="Green" title="E13">A1</font></span>:  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c1">x</font> = 
 <a href="sin_cos.html#K30" title="SIN_COS:func.30">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K30" title="SIN_COS:func.30">exp_R</a> 1</span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c1">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="taylor_1.html#T8" target="_self" title="TAYLOR_1:th.8">Th8</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="sin_cos.html#K30" title="SIN_COS:func.30">exp_R</a> 1</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c1">x</font> &amp; <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c1">x</font> &amp; <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c1">x</font> )</span><br/>

<a NAME="E2:13"/>
 <a href="sin_cos.html#K30" title="SIN_COS:func.30">exp_R</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T60" title="SIN_COS:th.60">SIN_COS:60</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:13"/>
( <span class="p1">(<span class="default"><a href="sin_cos.html#K30" title="SIN_COS:func.30">exp_R</a> 1</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c1">x</font> &amp; <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c1">x</font> &amp; <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="irrat_1.html#D7" title="IRRAT_1:def.7">IRRAT_1:def 7</a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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