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<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<a NAME="E1:89"/><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>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> ] <b>means </b> ex <font color="Olive">f</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 <br/>( <font color="Olive">f</font> <a href="funct_2.html#R1">=</a>  <a href="polynom5.html#K8" target="_self">FPower</a> <font color="Maroon">x</font>,<font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">f</font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a>  );<br/>
<a NAME="E2:89"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">f</font> =  <a href="polynom5.html#K8" target="_self">FPower</a> <font color="Maroon">x</font>,0 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/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E2:89_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a>  <a href="polynom5.html#K8" target="_self">FPower</a> <font color="Maroon">x</font>,0
 ;<br/>

<a NAME="E3:89_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="finsop_1.html#K3">--&gt;</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:89_1"/>
<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><b>end;</b></div>
<a NAME="E3:89"/><i><font color="Green">E38</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="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><b>proof </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>given </b><font color="Maroon">f</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><a NAME="E1:89_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a>  <a href="polynom5.html#K8" target="_self">FPower</a> <font color="Maroon">x</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:89_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">g</font> =  <a href="polynom5.html#K8" target="_self">FPower</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</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/>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E4:89_2"/>
<font color="Maroon">g</font> <a href="funct_2.html#R1">=</a>  <a href="polynom5.html#K8" target="_self">FPower</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">f1</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="E6:89_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#R1">=</a>  <a href="polynom5.html#K8" target="_self">FPower</a> <font color="Maroon">x</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E7:89_2"/><i><font color="Green">E46</font></i>: 
 <a href="polynom5.html#K8" target="_self">FPower</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="funct_2.html#R1">=</a> <font color="Maroon">f1</font> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K2">COMPLEX</a> </span>)</span>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E8:89_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#T5" target="_self">Th5</a>, , <a class="ref" href="cfcont_1.html#T65">CFCONT_1:65</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:89"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


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