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<b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">w</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">l</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K12" target="_self">Expan_e</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K12" target="_self">Expan_e</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">l</font>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K12" target="_self">Expan_e</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="comseq_3.html#T35">COMSEQ_3:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K12" target="_self">Expan_e</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="comseq_3.html#T27">COMSEQ_3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K12" target="_self">Expan_e</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
;<br/><br/>


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