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<b>let </b><font color="Maroon" title="c1">x</font> be   <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> ;<br/>

<a NAME="E1:89"/><i><font color="Green" title="E62">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>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> ] <b>means </b> ex <font color="Olive" title="b1">f</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 <br/>( <font color="Olive" title="b1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <font color="Maroon" title="c1">x</font>,$1 &amp; <font color="Olive" title="b1">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>  );<br/>
<a NAME="E2:89"/><i><font color="Green" title="E63">A2</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" title="c2">f</font> =  <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <font color="Maroon" title="c1">x</font>,0 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#E1:89"><i><font color="Green" title="E62">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">f</font>
;<br/>

<b>thus </b><a NAME="E2:89_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <font color="Maroon" title="c1">x</font>,0
 ;<br/>

<a NAME="E3:89_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</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>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="ref" href="polynom5.html#T67" target="_self" title="POLYNOM5:th.67">Th67</a></i>;<br/>
<b>hence </b><a NAME="E4:89_1"/>
<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#E1:89"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="polynom5.html#T64" target="_self" title="POLYNOM5:th.64">Th64</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:89"/><i><font color="Green" title="E64">A3</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c2">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>given </b><font color="Maroon" title="c3">f</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><a NAME="E1:89_2"/><i><font color="Green" title="E65">A4</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E2:89_2"/><i><font color="Green" title="E66">A5</font></i>: 
<font color="Maroon" title="c3">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> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">g</font> =  <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="txt" href="polynom5.html#E1:89"><i><font color="Green" title="E62">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">g</font>
;<br/>

<b>thus </b><a NAME="E4:89_2"/>
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">f1</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="E6:89_2"/><i><font color="Green" title="E67">A6</font></i>: 
<font color="Maroon" title="c5">f1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E7:89_2"/><i><font color="Green" title="E68">A7</font></i>: 
 <a href="polynom5.html#K8" title="POLYNOM5:func.8">FPower</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> </span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#T70" target="_self" title="POLYNOM5:th.70">Th70</a></i>;<br/>
<b>thus </b><a NAME="E8:89_2"/>
<font color="Maroon" title="c4">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#E1:89_2"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="polynom5.html#E2:89_2"><i><font color="Green" title="E66">A5</font></i></a>, <a class="txt" href="polynom5.html#E6:89_2"><i><font color="Green" title="E67">A6</font></i></a>, <a class="txt" href="polynom5.html#E7:89_2"><i><font color="Green" title="E68">A7</font></i></a>, <a class="ref" href="polynom5.html#T63" target="_self" title="POLYNOM5:th.63">Th63</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>
<b>thus </b><a NAME="E4:89"/>
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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</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#E2:89"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="polynom5.html#E3:89"><i><font color="Green" title="E64">A3</font></i></a>);</i><br/>


</div>
