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<div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <br/>  for <font color="Olive" title="b2">Fi</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <font color="Olive" title="b1">F</font> holds <br/> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b2">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Olive" title="b1">F</font></span>)</span>;<br/>
<a NAME="E1:18"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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">  for <font color="Olive" title="b1">Fi</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <font color="Maroon" title="c1">F</font> holds <br/> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b1">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c1">F</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Fi</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <font color="Maroon" title="c1">F</font> implies  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c2">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c1">F</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:18_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <font color="Maroon" title="c1">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c2">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c1">F</font></span>)</span></span><br/>

<span class="kw">then </span> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">Fi</font> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E3:18_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">Fi</font> <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="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">thus </span> <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c1">F</font></span>)</span> = 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="integr16.html#E10"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a>, <a class="ref" href="finseq_1.html#T20" title="FINSEQ_1:th.20">FINSEQ_1:20</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c2">Fi</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="complex1.html#T4" title="COMPLEX1:th.4">COMPLEX1:4</a>, <a class="ref" href="finseq_1.html#T20" title="FINSEQ_1:th.20">FINSEQ_1:20</a>, <a class="ref" href="rvsum_1.html#T72" title="RVSUM_1:th.72">RVSUM_1:72</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>
<div><span class="lab"><font color="Green" title="E14">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:18_2"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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">  for <font color="Olive" title="b1">Fi</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Olive" title="b1">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <font color="Maroon" title="c2">F</font> holds <br/> <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b1">Fi</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">Fi</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c3">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <font color="Maroon" title="c2">F</font> implies  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c3">Fi</font> )</span><br/><span class="kw">assume </span><a NAME="E1:18_2_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c3">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <font color="Maroon" title="c2">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c3">Fi</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c4">F1</font> = <font color="Maroon" title="c2">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">k</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  ;<br/><a NAME="E3:18_2_1"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">F1i</font> =  <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <font color="Maroon" title="c4">F1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></span>;<br/><a NAME="E6:18_2_1"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="finseq_3.html#T55" title="FINSEQ_3:th.55">FINSEQ_3:55</a></span>;<br/><span class="kw">hence </span> <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c2">F</font></span>)</span> = 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c6">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="integr16.html#E11"><span class="lab"><font color="Green" title="E8">Lm3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c4">F1</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c6">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T8" title="COMPLEX1:th.8">COMPLEX1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c5">F1i</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c6">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E3:18_2_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F1i</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T74" title="RVSUM_1:th.74">RVSUM_1:74</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c3">Fi</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E6:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="integr16.html#T6" target="_self" title="INTEGR16:th.6">Th6</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/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:18_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 ; <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="E3:18"/><span class="lab"><font color="Green" title="E15">A9</font></span>: 
 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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E14">A4</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c1">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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">  for <font color="Olive" title="b1">Fi</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <font color="Maroon" title="c1">F</font> holds <br/> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b1">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c1">F</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Fi</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <font color="Maroon" title="c1">F</font> implies  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c2">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c1">F</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E4:18"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
<font color="Maroon" title="c2">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <font color="Maroon" title="c1">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c2">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c1">F</font></span>)</span></span><br/>

<a NAME="E5:18"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> is    <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> 
 
;<br/>
<span class="kw">hence </span><a NAME="E6:18"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c2">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c1">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E16">A10</font></span></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>
