<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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/>

<i><font color="Green" title="E12">A1</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#D1" target="_self" title="POLYNOM5:def.1">Def1</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E2:12"/><i><font color="Green" title="E13">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>,<span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:12_2"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1"/>
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="polynom5.html#E1:12_2"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:12_2_1_1"/><b>then </b><i><font color="Green" title="E16">A5</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom5.html#E1:12_2"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></i>;<br/><a NAME="E3:12_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom5.html#D1" target="_self" title="POLYNOM5:def.1">Def1</a></i>;<br/><a NAME="E4:12_2_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span>
 <b>by </b><i><a class="txt" href="polynom5.html#E2:12_2_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><b>then </b><span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 = 
<span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="txt" href="polynom5.html#E2:12_2_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="polynom5.html#D1" target="_self" title="POLYNOM5:def.1">Def1</a></i>
;<br/><a NAME="E6:12_2_1_1"/><b>then </b>
<span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><b>hence </b><a NAME="E7:12_2_1_1"/>
<span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>,<span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E1:12_2_1_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_2"/><i><font color="Green" title="E15">A6</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:12_2_1_2"/><b>then </b><i><font color="Green" title="E16">A7</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom5.html#E1:12_2"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></i>;<br/><a NAME="E3:12_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom5.html#D1" target="_self" title="POLYNOM5:def.1">Def1</a></i>;<br/><a NAME="E4:12_2_1_2"/><b>then </b>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span>
 <b>by </b><i><a class="txt" href="polynom5.html#E2:12_2_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><b>then </b><span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 2 = 
<span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="txt" href="polynom5.html#E2:12_2_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="polynom5.html#D1" target="_self" title="POLYNOM5:def.1">Def1</a></i>
;<br/><a NAME="E6:12_2_1_2"/><b>then </b>
<span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><b>hence </b><a NAME="E7:12_2_1_2"/>
<span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>,<span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E1:12_2_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:12"/>
<span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>,<span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom5.html#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="polynom5.html#E2:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/>


</div>
