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<b>let </b><font color="Maroon">s</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">z</font> be  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b><font color="Maroon">z</font> <a href="newton.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font>;<br/>
<a NAME="E1:12"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/>
<font color="Maroon">z</font> <a href="newton.html#K2" target="_self">|^</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="rvsum_1.html#K17">Product</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">z</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T73">FINSEQ_2:73</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T125">RVSUM_1:125</a></i>
;<br/>
<b>hence </b><a NAME="E3:12_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:12"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">s</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon">z</font> <a href="newton.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2" target="_self">|^</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/>
<a NAME="E3:12_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<font color="Maroon">z</font> <a href="newton.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
 <a href="rvsum_1.html#K17">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="rvsum_1.html#K18">|-&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">z</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_2.html#T74">FINSEQ_2:74</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K17">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="rvsum_1.html#K18">|-&gt;</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T126">RVSUM_1:126</a></i>
;<br/>
<b>hence </b><a NAME="E5:12_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:12"/>
for <font color="Olive">s</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:12"/>
<font color="Maroon">z</font> <a href="newton.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2" target="_self">|^</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font>
 ;<br/>


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