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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">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="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span><br/><span class="kw">thus </span><a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">k</font> = 
1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="irrat_1.html#D5" target="_self" title="IRRAT_1:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D5" title="SIN_COS:def.5">SIN_COS:def 5</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:30"/>
 <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T45" title="SIN_COS:th.45">SIN_COS:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> 1</span><br/>

<span class="kw">thus </span> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> 1 = 
<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D23" title="SIN_COS:def.23">SIN_COS:def 23</a></span>
<br/>.= 
 <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="sin_cos.html#D22" title="SIN_COS:def.22">SIN_COS:def 22</a></span>
; <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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