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<b>let </b><font color="Maroon" title="c1">z</font>, <font color="Maroon" title="c2">w</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><a href="sin_cos.html#K12" title="SIN_COS:func.12">Expan</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>



<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D8" target="_self" title="SIN_COS:def.8">Def8</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="sin_cos.html#T1" target="_self" title="SIN_COS:th.1">Th1</a>, <a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="sin_cos.html#K12" title="SIN_COS:func.12">Expan</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E2:31"/><i><font color="Green" title="E24">A1</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="sin_cos.html#K12" title="SIN_COS:func.12">Expan</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D13" target="_self" title="SIN_COS:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E2:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="sin_cos.html#D10" target="_self" title="SIN_COS:def.10">Def10</a>, <a class="ref" href="sin_cos.html#T1" target="_self" title="SIN_COS:th.1">Th1</a></i>
<br/>.= 
<a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="comseq_3.html#K1" title="COMSEQ_3:func.1">GeoSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></i>
;<br/> <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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