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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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>   ex <font color="Olive" title="b2">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="b2">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="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b2">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>  )</span><br/>

<span class="kw">defpred </span><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> ] <span class="kw">means </span> 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="E1:89"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
 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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></span>;<br/>
<a NAME="E2:89"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
 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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">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#K2" title="NAT_1:func.2">+</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E62">A1</font></span></a></span>;<br/>
<span class="kw">given </span><font color="Maroon" title="c4">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><span class="kw"> such that </span><a NAME="E3:89_1"/><span class="lab"><font color="Green" title="E64">A3</font></span>: 
( <font color="Maroon" title="c4">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>) &amp; <font color="Maroon" title="c4">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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c3">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) &amp; <font color="Maroon" title="c3">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>  )</span><br/>

<span class="kw">thus </span><a NAME="E4:89_1"/>
<font color="Maroon" title="c3">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">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> </span><br/>

<a NAME="E5:89_1"/>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">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>) &amp;  <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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">f1</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T69" target="_self" title="POLYNOM5:th.69">Th70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:89_1"/>
<font color="Maroon" title="c3">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89_1"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="polynom5.html#T62" target="_self" title="POLYNOM5:th.62">Th63</a>, <a class="ref" href="cfcont_1.html#T43" title="CFCONT_1:th.43">CFCONT_1:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:89"/><span class="lab"><font color="Green" title="E64">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) 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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E62">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) &amp; <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>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:89_2"/>
<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">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> </span><br/>

<a NAME="E3:89_2"/>
<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="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T66" target="_self" title="POLYNOM5:th.66">Th67</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:89_2"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="ref" href="polynom5.html#T63" target="_self" title="POLYNOM5:th.63">Th64</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E3:89"><span class="lab"><font color="Green" title="E64">A4</font></span></a>, <a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E63">A2</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
