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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">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#K9" title="SIN_COS:func.9">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/>



<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">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="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D4" target="_self" title="SIN_COS:def.4">Def8</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><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></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">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="E2:28"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">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#K6" title="SIN_COS:func.6">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D9" target="_self" title="SIN_COS:def.9">Def13</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="sin_cos.html#D6" target="_self" title="SIN_COS:def.6">Def10</a>, <a class="ref" href="sin_cos.html#T1" target="_self" title="SIN_COS:th.1">Th1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></span>
;<br/> <a class="txt" 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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