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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K13" title="SIN_COS:func.13">Expan_e</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K13" title="SIN_COS:func.13">Expan_e</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><br/>





<div><span class="lab"><font color="Green" title="E26">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">l</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> implies <span class="p1">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<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> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" title="SIN_COS:func.13">Expan_e</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font> )</span><br/><span class="kw">assume </span><a NAME="E1:34_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c4">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ; <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#K14" title="SIN_COS:func.14">Alfa</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<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> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" title="SIN_COS:func.13">Expan_e</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<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> <font color="Maroon" title="c4">l</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <font color="Maroon" title="c3">k</font>,<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> <font color="Maroon" title="c4">l</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K13" title="SIN_COS:func.13">Expan_e</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<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> <font color="Maroon" title="c4">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="sin_cos.html#T12" target="_self" title="SIN_COS:th.12">Th12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" title="SIN_COS:func.13">Expan_e</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</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/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">k</font> = 
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K13" title="SIN_COS:func.13">Expan_e</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="comseq_3.html#T35" title="COMSEQ_3:th.35">COMSEQ_3:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K13" title="SIN_COS:func.13">Expan_e</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T27" title="COMSEQ_3:th.27">COMSEQ_3:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K14" title="SIN_COS:func.14">Alfa</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K13" title="SIN_COS:func.13">Expan_e</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</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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