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<b>let </b><font color="Maroon">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<a NAME="E1:91"/><i><font color="Green">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K2">COMPLEX</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> =  <a href="polynom5.html#K9" target="_self">Polynomial-Function</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">p</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a>  ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E3:91"/>
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a>  <a href="polynom5.html#K9" target="_self">Polynomial-Function</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">p</font>
 ;<br/>

<b>set </b><font color="Maroon">FuFF</font> =  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K2">COMPLEX</a> ,<a href="numbers.html#K2">COMPLEX</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K2">COMPLEX</a> ,<a href="numbers.html#K2">COMPLEX</a> ,   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K2">COMPLEX</a> ,<a href="numbers.html#K2">COMPLEX</a> ) <b>-&gt; </b>   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K2">COMPLEX</a> ,<a href="numbers.html#K2">COMPLEX</a>  =  <a href="funcsdom.html#K3">@</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">fadd</font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K2">COMPLEX</a> ,<a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E5:91"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K2">COMPLEX</a> ,<a href="numbers.html#K2">COMPLEX</a>  holds  <font color="Maroon">fadd</font> <a href="funcsdom.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">fzero</font> = <a href="numbers.html#K2">COMPLEX</a>  <a href="finsop_1.html#K3">--&gt;</a> <a href="complex1.html#K5">0c</a>  as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K2">COMPLEX</a> ,<a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T12">FUNCT_2:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">L</font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K2">COMPLEX</a> ,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<font color="Maroon">fadd</font>,<font color="Maroon">fzero</font> #) as  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L1">LoopStr</a>  ;<br/>
<div><i><font color="Green">E38</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">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u</font>, <font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><a NAME="E2:91_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">u1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K2">COMPLEX</a> ,<a href="numbers.html#K2">COMPLEX</a>  &amp; <font color="Maroon">v1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K2">COMPLEX</a> ,<a href="numbers.html#K2">COMPLEX</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#T12">FUNCT_2:12</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u</font>,<font color="Maroon">v</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K3">@</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">v1</font></span>)</span></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">v1</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="funcsdom.html#D1">FUNCSDOM:def 1</a></i>
<br/>.= 
 <a href="funcsdom.html#K3">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">v1</font></span>)</span> <a href="comseq_1.html#K2">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">v1</font></span>)</span> <a href="comseq_1.html#K2">+</a> <font color="Maroon">w1</font>
<b>by </b><i><a class="ref" href="funcsdom.html#D1">FUNCSDOM:def 1</a></i>
<br/>.= 
<font color="Maroon">u1</font> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i><a class="ref" href="cfunct_1.html#T22">CFUNCT_1:22</a></i>
<br/>.= 
 <a href="funcsdom.html#K3">@</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">w1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcsdom.html#D1">FUNCSDOM:def 1</a></i>
<br/>.= 
<font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="funcsdom.html#K3">@</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">w1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcsdom.html#D1">FUNCSDOM:def 1</a></i>
<br/>.= 
<font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E9:91"/><i><font color="Green">E40</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fzero</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>;<br/>
<div><i><font color="Green">E45</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">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><div><i><font color="Green">E46</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">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">fzero</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">fzero</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <a href="complex1.html#K5">0c</a> 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>

;<br/></div><b>end;</b></div><b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> = 
<font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
 <a href="funcsdom.html#K3">@</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">fzero</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">v1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">fzero</font>
<b>by </b><i><a class="ref" href="funcsdom.html#D1">FUNCSDOM:def 1</a></i>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="polynom5.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_2.html#T113">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">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><b>reconsider </b><font color="Maroon">w</font> =  <a href="comseq_1.html#K5">-</a> <font color="Maroon">v1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="funct_2.html#T12">FUNCT_2:12</a></i>;<br/><b>take </b><font color="Maroon">w</font> = <font color="Maroon">w</font>;<br/><div><i><font color="Green">E47</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">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="comseq_1.html#K5">-</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_1.html#K5">-</a> <font color="Maroon">v1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9">COMSEQ_1:def 9</a></i>
<br/>.= 
<font color="Maroon">fzero</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/></div><b>end;</b></div><b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
 <a href="funcsdom.html#K3">@</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="comseq_1.html#K5">-</a> <font color="Maroon">v1</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v1</font> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="comseq_1.html#K5">-</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="funcsdom.html#D1">FUNCSDOM:def 1</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i>, <a class="ref" href="polynom5.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>
;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a>  <a href="rlvect_1.html#L1">LoopStr</a>  <b>by </b><i>, <a class="ref" href="polynom5.html#T5" target="_self">Th5</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a>, <a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom5.html#K8" target="_self">FPower</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span>;<br/>
<div><i><font color="Green">E48</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:91_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">x</font> =  <a href="polynom5.html#K8" target="_self">FPower</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T12">FUNCT_2:12</a></i>;<br/><b>take </b><font color="Maroon">x</font> = <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E3:91_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E15:91"/><i><font color="Green">E49</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 <b>and </b><br/><a NAME="E16:91"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="ref" href="polynom5.html#T6" target="_self">Th6</a>);<br/></i>
<a NAME="E17:91"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">g</font> being  <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a>   st <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/><font color="Olive">g</font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a> ;<br/>
<a NAME="E18:91"/><i><font color="Green">E59</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">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:91_5"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> 0</span>)</span>
 ;<br/>

<a NAME="E2:91_5"/>
<font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> 0 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<b>then </b><font color="Maroon">g</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom5.html#T8" target="_self">Th8</a>, <a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>
<br/>.= 
<a href="numbers.html#K2">COMPLEX</a>  <a href="finsop_1.html#K3">--&gt;</a> <a href="complex1.html#K5">0c</a> 
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E4:91_5"/>
<font color="Maroon">g</font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E19:91"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:91_6"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">g</font> being  <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a>   st <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> holds <br/><font color="Olive">g</font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a> 
 ;<br/>

<b>let </b><font color="Maroon">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E2:91_6"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">g1</font> =  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E4:91_6"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">g1</font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a> 
 
<b>by </b><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">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">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">E65</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:91_6_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:91_6_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:91_6_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E130</font></i>: <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="polynom5.html#K8" target="_self">FPower</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="polynom5.html#K8" target="_self">FPower</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
 <a href="polynom5.html#K8" target="_self">FPower</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><b>consider </b><font color="Maroon">g2</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E7:91_6_1_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">g2</font> <a href="funct_2.html#R1">=</a>  <a href="polynom5.html#K8" target="_self">FPower</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E8:91_6_1_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">g2</font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>by </b><i><a class="txt" href="polynom5.html#E19:91"><i><font color="Green">E61</font></i></a></i>;<br/><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="polynom3.html#T19">POLYNOM3:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>then </b><font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#D1" target="_self">Def1</a>, <a class="ref" href="fvsum_1.html#T87">FVSUM_1:87</a></i>
<br/>.= 
<font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
 <a href="funcsdom.html#K3">@</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">g2</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E1:91_6_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="polynom5.html#T9" target="_self">Th9</a>, <a class="ref" href="polynom5.html#T10" target="_self">Th10</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">g2</font>
<b>by </b><i><a class="ref" href="funcsdom.html#D1">FUNCSDOM:def 1</a></i>
;<br/><b>hence </b><a NAME="E11:91_6_1_1"/>
<font color="Maroon">g</font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>by </b><i>, <a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="cfcont_1.html#T65">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">E133</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:91_6_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:91_6_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then </b><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
;<br/><b>hence </b><a NAME="E5:91_6_1_2"/>
<font color="Maroon">g</font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>by </b><i>, <a class="ref" href="polynom5.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E20:91"/><i><font color="Green">E134</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="polynom5.html#T8" target="_self">Th8</a>);<br/></i>
<b>reconsider </b><font color="Maroon">SF</font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121">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">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><b>consider </b><font color="Maroon">H</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E3:91_7"/><i><font color="Green">E135</font></i>: 
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E4:91_7"/><i><font color="Green">E136</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E5:91_7"/><i><font color="Green">E137</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom4.html#D2">POLYNOM4:def 2</a></i>;<br/><a NAME="E6:91_7"/><i><font color="Green">E138</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">SFk</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a>   st <font color="Olive">SFk</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/> <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">SFk</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>;<br/><a NAME="E7:91_7"/><i><font color="Green">E141</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">SFk</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:91_7_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">SFk</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> 0</span>)</span>
 ;<br/>

<a NAME="E2:91_7_1"/>
<font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> 0 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<b>then </b><i><font color="Green">E143</font></i>: <font color="Maroon">SFk</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom5.html#T9" target="_self">Th9</a>, <a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>
<br/>.= 
<a href="numbers.html#K2">COMPLEX</a>  <a href="finsop_1.html#K3">--&gt;</a> <a href="complex1.html#K5">0c</a> 
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
;<br/>
<a NAME="E4:91_7_1"/>
<font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> 0 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> 
 
;<br/>
<b>hence </b> <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> 0</span>)</span> = 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i><a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>
<br/>.= 
<font color="Maroon">SFk</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="polynom5.html#T10" target="_self">Th10</a>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E8:91_7"/><i><font color="Green">E144</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:91_7_2"/><i><font color="Green">E145</font></i>: 
for <font color="Olive">SFk</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a>   st <font color="Olive">SFk</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> holds <br/> <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">SFk</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/>

<b>let </b><font color="Maroon">SFk</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E2:91_7_2"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">SFk</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">SFk1</font> =  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121">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">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">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">E147</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:91_7_2_1_1"/><b>then </b><i><font color="Green">E148</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:91_7_2_1_1"/><i><font color="Green">E149</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:91_7_2_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:91_7_2_1_1"/><b>then </b><i><font color="Green">E150</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E151</font></i>: <font color="Maroon">H</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom5.html#K8" target="_self">FPower</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/><i><font color="Green">E153</font></i>: <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="polynom5.html#K8" target="_self">FPower</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="polynom5.html#K8" target="_self">FPower</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
 <a href="polynom5.html#K8" target="_self">FPower</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><b>reconsider </b><font color="Maroon">g2</font> =  <a href="polynom5.html#K8" target="_self">FPower</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">k</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i/>;<br/><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T15" target="_self">Th15</a>, <a class="ref" href="polynom3.html#T19">POLYNOM3:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>then </b><i><font color="Green">E154</font></i>: <font color="Maroon">SFk</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="fvsum_1.html#T87">FVSUM_1:87</a></i>
<br/>.= 
<font color="Maroon">fadd</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
 <a href="funcsdom.html#K3">@</a> <span class="p1">(<span class="default"><font color="Maroon">SFk1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">g2</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">SFk1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">g2</font>
<b>by </b><i><a class="ref" href="funcsdom.html#D1">FUNCSDOM:def 1</a></i>
;<br/><a NAME="E11:91_7_2_1_1"/><i><font color="Green">E155</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">SFk1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="polynom5.html#T10" target="_self">Th10</a></i>;<br/><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i>, , <a class="ref" href="polynom5.html#T15" target="_self">Th15</a>, <a class="ref" href="polynom3.html#T19">POLYNOM3:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b> <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T87">FVSUM_1:87</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">SFk1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T16" target="_self">Th16</a>, <a class="ref" href="polynom5.html#T21" target="_self">Th21</a>, <a class="ref" href="complfld.html#T3">COMPLFLD:3</a></i>
<br/>.= 
<font color="Maroon">SFk</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="polynom5.html#T20" target="_self">Th20</a>, <a class="ref" href="comseq_1.html#D4">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">E156</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:91_7_2_1_2"/><b>then </b><i><font color="Green">E157</font></i>: 
( <font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><a NAME="E3:91_7_2_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:91_7_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="polynom5.html#T15" target="_self">Th15</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:91_7_2_1_2"/><b>then </b>
( <font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><b>hence </b><a NAME="E6:91_7_2_1_2"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">SFk</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="polynom5.html#T10" target="_self">Th10</a>, <a class="ref" href="polynom5.html#T13" target="_self">Th13</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">E158</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="polynom5.html#T9" target="_self">Th9</a>);<br/></i><a NAME="E10:91_7"/><i><font color="Green">E159</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">SF</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><b>thus </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">H</font>
<b>by </b><i><a class="ref" href="polynom5.html#D1" target="_self">Def1</a>, </i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
<br/>.= 
<font color="Maroon">SF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="polynom5.html#T10" target="_self">Th10</a>, <a class="ref" href="polynom5.html#T13" target="_self">Th13</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E23:91"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">SF</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E24:91"/>
<font color="Maroon">f</font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>by </b><i>, </i>;<br/>


</div>
