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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>

<a NAME="E1:91"/><i><font color="Green" title="E64">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">f</font> =  <a href="polynom5.html#K9" title="POLYNOM5:func.9">Polynomial-Function</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">p</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c2">f</font>
;<br/>

<b>thus </b><a NAME="E3:91"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="polynom5.html#K9" title="POLYNOM5:func.9">Polynomial-Function</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">p</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">FuFF</font> =  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ) <b>-&gt; </b>   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  =  <a href="funcsdom.html#K3" title="FUNCSDOM:func.3">@</a> <span class="p1">(<span class="default">$1 <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> $2</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c4">fadd</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E5:91"/><i><font color="Green" title="E65">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  holds  <font color="Maroon" title="c4">fadd</font> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">fzero</font> = <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">L</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> </span>)</span>,<font color="Maroon" title="c4">fadd</font>,<font color="Maroon" title="c5">fzero</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a>  ;<br/>
<div><i><font color="Green" title="E66">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">u</font>, <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c9">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">L</font>;<br/><b>reconsider </b><font color="Maroon" title="c10">u1</font> = <font color="Maroon" title="c7">u</font>, <font color="Maroon" title="c11">v1</font> = <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c12">w1</font> = <font color="Maroon" title="c9">w</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/><a NAME="E2:91_1"/><i><font color="Green" title="E67">A4</font></i>: 
( <font color="Maroon" title="c10">u1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c11">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  &amp; <font color="Maroon" title="c11">v1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c12">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c7">u</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c8">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c9">w</font> = 
<font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">u</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c8">v</font></span>)</span>,<font color="Maroon" title="c9">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">v</font></span>)</span>,<font color="Maroon" title="c9">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K3" title="FUNCSDOM:func.3">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">u1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c11">v1</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">w</font>
<b>by </b><i><a class="txt" href="polynom5.html#E5:91"><i><font color="Green" title="E65">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">u1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c11">v1</font></span>)</span>,<font color="Maroon" title="c9">w</font>
<b>by </b><i><a class="ref" href="funcsdom.html#D1" title="FUNCSDOM:def.1">FUNCSDOM:def 1</a></i>
<br/>.= 
 <a href="funcsdom.html#K3" title="FUNCSDOM:func.3">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">u1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c11">v1</font></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c12">w1</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E5:91"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="polynom5.html#E2:91_1"><i><font color="Green" title="E67">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">u1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c11">v1</font></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c12">w1</font>
<b>by </b><i><a class="ref" href="funcsdom.html#D1" title="FUNCSDOM:def.1">FUNCSDOM:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c10">u1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">v1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c12">w1</font></span>)</span>
<b>by </b><i><a class="ref" href="cfunct_1.html#T22" title="CFUNCT_1:th.22">CFUNCT_1:22</a></i>
<br/>.= 
 <a href="funcsdom.html#K3" title="FUNCSDOM:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">u1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">v1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c12">w1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcsdom.html#D1" title="FUNCSDOM:def.1">FUNCSDOM:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">u</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">v1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c12">w1</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E5:91"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="polynom5.html#E2:91_1"><i><font color="Green" title="E67">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">u</font>,<span class="p1">(<span class="default"><a href="funcsdom.html#K3" title="FUNCSDOM:func.3">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">v1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c12">w1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcsdom.html#D1" title="FUNCSDOM:def.1">FUNCSDOM:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">u</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">w</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E5:91"><i><font color="Green" title="E65">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">u</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c9">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon" title="c7">u</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c9">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E67">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">L</font>;<br/><b>reconsider </b><font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c7">v</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/><div><i><font color="Green" title="E68">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</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> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c5">fzero</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fzero</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c8">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>

;<br/></div><b>end;</b></div><b>thus </b><font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c6">L</font></span>)</span> = 
<font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">v</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c6">L</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></i>
<br/>.= 
 <a href="funcsdom.html#K3" title="FUNCSDOM:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c5">fzero</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E5:91"><i><font color="Green" title="E65">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">v1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c5">fzero</font>
<b>by </b><i><a class="ref" href="funcsdom.html#D1" title="FUNCSDOM:def.1">FUNCSDOM:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c7">v</font>
<b>by </b><i><a class="txt" href="polynom5.html#E2:91_2"><i><font color="Green" title="E68">A6</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">L</font>;<br/><b>reconsider </b><font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c7">v</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">w</font> =  <a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <font color="Maroon" title="c8">v1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">L</font> <b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/><b>take </b><font color="Maroon" title="c10">w</font> = <font color="Maroon" title="c9">w</font>;<br/><div><i><font color="Green" title="E68">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</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> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p2">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <font color="Maroon" title="c8">v1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</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_1.html#K5" title="COMSEQ_1:func.5">-</a> <font color="Maroon" title="c8">v1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9" title="COMSEQ_1:def.9">COMSEQ_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c5">fzero</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/></div><b>end;</b></div><b>thus </b><font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c10">w</font> = 
<font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c10">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></i>
<br/>.= 
 <a href="funcsdom.html#K3" title="FUNCSDOM:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p2">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <font color="Maroon" title="c8">v1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E5:91"><i><font color="Green" title="E65">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">v1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p1">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <font color="Maroon" title="c8">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="funcsdom.html#D1" title="FUNCSDOM:def.1">FUNCSDOM:def 1</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c6">L</font>
<b>by </b><i><a class="txt" href="polynom5.html#E3:91_3"><i><font color="Green" title="E68">A7</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>
;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">L</font> = <font color="Maroon" title="c6">L</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a>  <b>by </b><i><a class="txt" href="polynom5.html#E8:91"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="polynom5.html#E9:91"><i><font color="Green" title="E67">A5</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a>, <a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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 href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>;<br/>
<div><i><font color="Green" title="E68">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">n</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> ;<br/><b>assume </b><a NAME="E1:91_4"/>
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c9">x</font> =  <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">L</font> <b>by </b><i><a class="txt" href="polynom5.html#E1:91"><i><font color="Green" title="E64">A1</font></i></a>, <a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/><b>take </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">x</font>;<br/><b>thus </b><a NAME="E3:91_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c10">x</font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">L</font><b> such that </b><br/><a NAME="E14:91"/><i><font color="Green" title="E69">A9</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>and </b><br/><a NAME="E15:91"/><i><font color="Green" title="E70">A10</font></i>: 
for <font color="Olive" title="b1">n</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>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="polynom5.html#E12:91"><i><font color="Green" title="E68">A8</font></i></a>);<br/></i>
<a NAME="E16:91"/><i><font color="Green" title="E71">A11</font></i>: 
<font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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> ] <b>means </b>for <font color="Olive" title="b1">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span> holds <br/><font color="Olive" title="b1">g</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>
<a NAME="E17:91"/><i><font color="Green" title="E72">A12</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:91_5"/><i><font color="Green" title="E73">A13</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 0</span>)</span>
 ;<br/>

<a NAME="E2:91_5"/>
<font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">L</font>
 
;<br/>
<b>then </b><font color="Maroon" title="c9">g</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c7">L</font>
<b>by </b><i><a class="txt" href="polynom5.html#E1:91_5"><i><font color="Green" title="E73">A13</font></i></a>, <a class="ref" href="rlvect_1.html#T60" title="RLVECT_1:th.60">RLVECT_1:60</a></i>
<br/>.= 
<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 

;<br/>
<b>hence </b><a NAME="E4:91_5"/>
<font color="Maroon" title="c9">g</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 <b>by </b><i><a class="ref" href="polynom5.html#T64" target="_self" title="POLYNOM5:th.64">Th64</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:91"/><i><font color="Green" title="E73">A14</font></i>: 
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>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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> ;<br/>

<b>assume </b><a NAME="E1:91_6"/><i><font color="Green" title="E74">A15</font></i>: 
for <font color="Olive" title="b1">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> holds <br/><font color="Olive" title="b1">g</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c10">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E2:91_6"/><i><font color="Green" title="E75">A16</font></i>: 
<font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">g1</font> =  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<a NAME="E4:91_6"/><i><font color="Green" title="E76">A17</font></i>: 
<font color="Maroon" title="c11">g1</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<b>by </b><i><a class="txt" href="polynom5.html#E1:91_6"><i><font color="Green" title="E74">A15</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_6_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_6_1_1"><b>suppose </b></a><a NAME="E1:91_6_1_1"/><i><font color="Green" title="E77">A18</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">k</font>
 ;<br/><div class="add"><a NAME="E2:91_6_1_1"/><b>then </b><i><font color="Green" title="E78">A19</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:91_6_1_1"/>
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E4:91_6_1_1"/><b>then </b><i><font color="Green" title="E79">A20</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c8">F</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E2:91_6_1_1"><i><font color="Green" title="E78">A19</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E80">A21</font></i>: <font color="Maroon" title="c8">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E14:91"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="polynom5.html#E15:91"><i><font color="Green" title="E70">A10</font></i></a>, <a class="txt" href="polynom5.html#E4:91_6_1_1"><i><font color="Green" title="E79">A20</font></i></a></i>
<br/>.= 
 <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
 <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font></span>)</span>,<font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
;<br/><b>consider </b><font color="Maroon" title="c12">g2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E7:91_6_1_1"/><i><font color="Green" title="E81">A22</font></i>: 
<font color="Maroon" title="c12">g2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font></span>)</span>,<font color="Maroon" title="c9">k</font>
 <b>and </b><br/><a NAME="E8:91_6_1_1"/><i><font color="Green" title="E82">A23</font></i>: 
<font color="Maroon" title="c12">g2</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 <b>by </b><i><a class="ref" href="polynom5.html#T71" target="_self" title="POLYNOM5:th.71">Th71</a></i>;<br/><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="polynom5.html#E1:91_6_1_1"><i><font color="Green" title="E77">A18</font></i></a>, <a class="ref" href="polynom3.html#T19" title="POLYNOM3:th.19">POLYNOM3:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
<b>by </b><i><a class="txt" href="polynom5.html#E4:91_6_1_1"><i><font color="Green" title="E79">A20</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>then </b><font color="Maroon" title="c10">g</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E2:91_6"><i><font color="Green" title="E75">A16</font></i></a>, <a class="ref" href="fvsum_1.html#T87" title="FVSUM_1:th.87">FVSUM_1:87</a></i>
<br/>.= 
<font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></i>
<br/>.= 
 <a href="funcsdom.html#K3" title="FUNCSDOM:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c12">g2</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E5:91"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="polynom5.html#E5:91_6_1_1"><i><font color="Green" title="E80">A21</font></i></a>, <a class="txt" href="polynom5.html#E7:91_6_1_1"><i><font color="Green" title="E81">A22</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">g1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c12">g2</font>
<b>by </b><i><a class="ref" href="funcsdom.html#D1" title="FUNCSDOM:def.1">FUNCSDOM:def 1</a></i>
;<br/><b>hence </b><a NAME="E11:91_6_1_1"/>
<font color="Maroon" title="c10">g</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 <b>by </b><i><a class="txt" href="polynom5.html#E4:91_6"><i><font color="Green" title="E76">A17</font></i></a>, <a class="txt" href="polynom5.html#E8:91_6_1_1"><i><font color="Green" title="E82">A23</font></i></a>, <a class="ref" href="cfcont_1.html#T65" title="CFCONT_1:th.65">CFCONT_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_6_1_2"><b>suppose </b></a><a NAME="E1:91_6_1_2"/><i><font color="Green" title="E77">A24</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 ;<br/><div class="add"><a NAME="E2:91_6_1_2"/>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:91_6_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="polynom5.html#E1:91_6_1_2"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then </b><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c8">F</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>
<br/>.= 
<font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="txt" href="polynom5.html#E1:91_6_1_2"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>
;<br/><b>hence </b><a NAME="E5:91_6_1_2"/>
<font color="Maroon" title="c10">g</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 <b>by </b><i><a class="txt" href="polynom5.html#E1:91_6"><i><font color="Green" title="E74">A15</font></i></a>, <a class="txt" href="polynom5.html#E2:91_6"><i><font color="Green" title="E75">A16</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E19:91"/><i><font color="Green" title="E74">A25</font></i>: 
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>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="polynom5.html#E17:91"><i><font color="Green" title="E72">A12</font></i></a>, <a class="txt" href="polynom5.html#E18:91"><i><font color="Green" title="E73">A14</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c9">SF</font> =  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c8">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</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> ;<br/><b>reconsider </b><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c10">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">H</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <b> such that </b><br/><a NAME="E3:91_7"/><i><font color="Green" title="E75">A26</font></i>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c12">H</font>
 <b>and </b><br/><a NAME="E4:91_7"/><i><font color="Green" title="E76">A27</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font>
 <b>and </b><br/><a NAME="E5:91_7"/><i><font color="Green" title="E77">A28</font></i>: 
for <font color="Olive" title="b1">n</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>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c12">H</font> holds <br/><font color="Maroon" title="c12">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">x1</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom4.html#D2" title="POLYNOM4:def.2">POLYNOM4:def 2</a></i>;<br/><a NAME="E6:91_7"/><i><font color="Green" title="E78">A29</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E14:91"><i><font color="Green" title="E69">A9</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <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> ] <b>means </b>for <font color="Olive" title="b1">SFk</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <font color="Olive" title="b1">SFk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span> holds <br/> <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">SFk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>;<br/><a NAME="E7:91_7"/><i><font color="Green" title="E79">A30</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">SFk</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:91_7_1"/><i><font color="Green" title="E80">A31</font></i>: 
<font color="Maroon" title="c13">SFk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 0</span>)</span>
 ;<br/>

<a NAME="E2:91_7_1"/>
<font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">L</font>
 
;<br/>
<b>then </b><i><font color="Green" title="E81">A32</font></i>: <font color="Maroon" title="c13">SFk</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c7">L</font>
<b>by </b><i><a class="txt" href="polynom5.html#E1:91_7_1"><i><font color="Green" title="E80">A31</font></i></a>, <a class="ref" href="rlvect_1.html#T60" title="RLVECT_1:th.60">RLVECT_1:60</a></i>
<br/>.= 
<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 

;<br/>
<a NAME="E4:91_7_1"/>
<font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 
;<br/>
<b>hence </b> <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 0</span>)</span> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<b>by </b><i><a class="ref" href="rlvect_1.html#T60" title="RLVECT_1:th.60">RLVECT_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c13">SFk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="polynom5.html#E3:91_7_1"><i><font color="Green" title="E81">A32</font></i></a>, <a class="ref" href="complfld.html#T9" title="COMPLFLD:th.9">COMPLFLD:9</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E8:91_7"/><i><font color="Green" title="E80">A33</font></i>: 
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>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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> ;<br/>

<b>assume </b><a NAME="E1:91_7_2"/><i><font color="Green" title="E81">A34</font></i>: 
for <font color="Olive" title="b1">SFk</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <font color="Olive" title="b1">SFk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">k</font></span>)</span> holds <br/> <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">SFk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c14">SFk</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E2:91_7_2"/><i><font color="Green" title="E82">A35</font></i>: 
<font color="Maroon" title="c14">SFk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c15">SFk1</font> =  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">k</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_7_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_7_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_7_2_1_1"><b>suppose </b></a><a NAME="E1:91_7_2_1_1"/><i><font color="Green" title="E83">A36</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">k</font>
 ;<br/><div class="add"><a NAME="E2:91_7_2_1_1"/><b>then </b><i><font color="Green" title="E84">A37</font></i>: 
<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:91_7_2_1_1"/><i><font color="Green" title="E85">A38</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c12">H</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E14:91"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="polynom5.html#E4:91_7"><i><font color="Green" title="E76">A27</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:91_7_2_1_1"/>
<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E5:91_7_2_1_1"/><b>then </b><i><font color="Green" title="E86">A39</font></i>: 
<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c8">F</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E2:91_7_2_1_1"><i><font color="Green" title="E84">A37</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E87">A40</font></i>: <font color="Maroon" title="c12">H</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c12">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E3:91_7_2_1_1"><i><font color="Green" title="E85">A38</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">x1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E5:91_7"><i><font color="Green" title="E77">A28</font></i></a>, <a class="txt" href="polynom5.html#E3:91_7_2_1_1"><i><font color="Green" title="E85">A38</font></i></a>, <a class="txt" href="polynom5.html#E5:91_7_2_1_1"><i><font color="Green" title="E86">A39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">x1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c13">k</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<font color="Maroon" title="c13">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="ref" href="polynom5.html#D11" target="_self" title="POLYNOM5:def.11">Def11</a></i>
;<br/><i><font color="Green" title="E88">A41</font></i>: <font color="Maroon" title="c8">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E5:91_7_2_1_1"><i><font color="Green" title="E86">A39</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E14:91"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="polynom5.html#E15:91"><i><font color="Green" title="E70">A10</font></i></a>, <a class="txt" href="polynom5.html#E5:91_7_2_1_1"><i><font color="Green" title="E86">A39</font></i></a></i>
<br/>.= 
 <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
 <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<font color="Maroon" title="c13">k</font>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c16">g2</font> =  <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<font color="Maroon" title="c13">k</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="txt" href="polynom5.html#E1:91"><i><font color="Green" title="E64">A1</font></i></a></i>;<br/><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="polynom5.html#E1:91_7_2_1_1"><i><font color="Green" title="E83">A36</font></i></a>, <a class="ref" href="polynom3.html#T19" title="POLYNOM3:th.19">POLYNOM3:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
<b>by </b><i><a class="txt" href="polynom5.html#E5:91_7_2_1_1"><i><font color="Green" title="E86">A39</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>then </b><i><font color="Green" title="E89">A42</font></i>: <font color="Maroon" title="c14">SFk</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E2:91_7_2"><i><font color="Green" title="E82">A35</font></i></a>, <a class="ref" href="fvsum_1.html#T87" title="FVSUM_1:th.87">FVSUM_1:87</a></i>
<br/>.= 
<font color="Maroon" title="c4">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></i>
<br/>.= 
 <a href="funcsdom.html#K3" title="FUNCSDOM:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">SFk1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c16">g2</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E5:91"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="polynom5.html#E7:91_7_2_1_1"><i><font color="Green" title="E88">A41</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c15">SFk1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c16">g2</font>
<b>by </b><i><a class="ref" href="funcsdom.html#D1" title="FUNCSDOM:def.1">FUNCSDOM:def 1</a></i>
;<br/><a NAME="E11:91_7_2_1_1"/><i><font color="Green" title="E90">A43</font></i>: 
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">SFk1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E1:91_7_2"><i><font color="Green" title="E81">A34</font></i></a></i>;<br/><font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="polynom5.html#E4:91_7"><i><font color="Green" title="E76">A27</font></i></a>, <a class="txt" href="polynom5.html#E6:91_7"><i><font color="Green" title="E78">A29</font></i></a>, <a class="txt" href="polynom5.html#E1:91_7_2_1_1"><i><font color="Green" title="E83">A36</font></i></a>, <a class="ref" href="polynom3.html#T19" title="POLYNOM3:th.19">POLYNOM3:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
<b>by </b><i><a class="txt" href="polynom5.html#E3:91_7_2_1_1"><i><font color="Green" title="E85">A38</font></i></a>, <a class="txt" href="polynom5.html#E5:91_7_2_1_1"><i><font color="Green" title="E86">A39</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b> <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T87" title="FVSUM_1:th.87">FVSUM_1:87</a></i>
<br/>.= 
<font color="Maroon" title="c14">SFk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="polynom5.html#E6:91_7_2_1_1"><i><font color="Green" title="E87">A40</font></i></a>, <a class="txt" href="polynom5.html#E11:91_7_2_1_1"><i><font color="Green" title="E90">A43</font></i></a>, <a class="txt" href="polynom5.html#E10:91_7_2_1_1"><i><font color="Green" title="E89">A42</font></i></a>, <a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_7_2_1_2"><b>suppose </b></a><a NAME="E1:91_7_2_1_2"/><i><font color="Green" title="E83">A44</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font>
 ;<br/><div class="add"><a NAME="E2:91_7_2_1_2"/><b>then </b><i><font color="Green" title="E84">A45</font></i>: 
( <font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> &amp; <font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">H</font> )
 <b>by </b><i><a class="txt" href="polynom5.html#E4:91_7"><i><font color="Green" title="E76">A27</font></i></a>, <a class="txt" href="polynom5.html#E6:91_7"><i><font color="Green" title="E78">A29</font></i></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/><a NAME="E3:91_7_2_1_2"/>
<font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E4:91_7_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="polynom5.html#E1:91_7_2_1_2"><i><font color="Green" title="E83">A44</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:91_7_2_1_2"/><b>then </b>
( <font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> &amp; <font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">H</font> )
 <b>by </b><i><a class="txt" href="polynom5.html#E4:91_7"><i><font color="Green" title="E76">A27</font></i></a>, <a class="txt" href="polynom5.html#E6:91_7"><i><font color="Green" title="E78">A29</font></i></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/><b>hence </b><a NAME="E6:91_7_2_1_2"/>
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">SFk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E1:91_7_2"><i><font color="Green" title="E81">A34</font></i></a>, <a class="txt" href="polynom5.html#E2:91_7_2"><i><font color="Green" title="E82">A35</font></i></a>, <a class="txt" href="polynom5.html#E2:91_7_2_1_2"><i><font color="Green" title="E84">A45</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E9:91_7"/><i><font color="Green" title="E81">A46</font></i>: 
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>3</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="polynom5.html#E7:91_7"><i><font color="Green" title="E79">A30</font></i></a>, <a class="txt" href="polynom5.html#E8:91_7"><i><font color="Green" title="E80">A33</font></i></a>);<br/></i><a NAME="E10:91_7"/><i><font color="Green" title="E82">A47</font></i>: 
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">SF</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/><b>thus </b><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> = 
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c12">H</font>
<b>by </b><i><a class="txt" href="polynom5.html#E3:91_7"><i><font color="Green" title="E75">A26</font></i></a>, <a class="ref" href="polynom5.html#D12" target="_self" title="POLYNOM5:def.12">Def12</a></i>
<br/>.= 
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>
<br/>.= 
<font color="Maroon" title="c9">SF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="polynom5.html#E4:91_7"><i><font color="Green" title="E76">A27</font></i></a>, <a class="txt" href="polynom5.html#E6:91_7"><i><font color="Green" title="E78">A29</font></i></a>, <a class="txt" href="polynom5.html#E9:91_7"><i><font color="Green" title="E81">A46</font></i></a>, <a class="txt" href="polynom5.html#E10:91_7"><i><font color="Green" title="E82">A47</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E22:91"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">SF</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E23:91"/>
<font color="Maroon" title="c2">f</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 <b>by </b><i><a class="txt" href="polynom5.html#E16:91"><i><font color="Green" title="E71">A11</font></i></a>, <a class="txt" href="polynom5.html#E19:91"><i><font color="Green" title="E74">A25</font></i></a></i>;<br/>


</div>
