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<div class="add">

<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> , <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> , <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<i><font color="Green">E33</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span>
<b>by </b><i/>
<br/>.= 
3
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E2:13"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">x</font></span><a href="complfld.html#K3">.|</a></span>,<span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">y</font></span><a href="complfld.html#K3">.|</a></span>,<span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 3
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </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>assume </b><a NAME="E1:13_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1"><b>suppose </b></a><a NAME="E1:13_2_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:13_2_1_1"/><i><font color="Green">E40</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="topreal3.html#T6">TOPREAL3:6</a></i>;<br/><a NAME="E3:13_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i/>;<br/><a NAME="E4:13_2_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>then </b><span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> 1 = 
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i>, </i>
;<br/><a NAME="E6:13_2_1_1"/><b>then </b>
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">x</font></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>;<br/><b>hence </b><a NAME="E7:13_2_1_1"/>
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">x</font></span><a href="complfld.html#K3">.|</a></span>,<span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">y</font></span><a href="complfld.html#K3">.|</a></span>,<span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_2"><b>suppose </b></a><a NAME="E1:13_2_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:13_2_1_2"/><i><font color="Green">E46</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="topreal3.html#T6">TOPREAL3:6</a></i>;<br/><a NAME="E3:13_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i/>;<br/><a NAME="E4:13_2_1_2"/><b>then </b>
2 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>then </b><span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> 2 = 
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i>, </i>
;<br/><a NAME="E6:13_2_1_2"/><b>then </b>
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">y</font></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>;<br/><b>hence </b><a NAME="E7:13_2_1_2"/>
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">x</font></span><a href="complfld.html#K3">.|</a></span>,<span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">y</font></span><a href="complfld.html#K3">.|</a></span>,<span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_3"><b>suppose </b></a><a NAME="E1:13_2_1_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:13_2_1_3"/><i><font color="Green">E48</font></i>: 
3 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="topreal3.html#T6">TOPREAL3:6</a></i>;<br/><a NAME="E3:13_2_1_3"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i/>;<br/><a NAME="E4:13_2_1_3"/><b>then </b>
3 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>then </b><span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> 3 = 
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="finseq_4.html#K4">/.</a> 3
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i>, </i>
;<br/><a NAME="E6:13_2_1_3"/><b>then </b>
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> 3 <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>;<br/><b>hence </b><a NAME="E7:13_2_1_3"/>
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">x</font></span><a href="complfld.html#K3">.|</a></span>,<span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">y</font></span><a href="complfld.html#K3">.|</a></span>,<span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:13"/>
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">x</font></span><a href="complfld.html#K3">.|</a></span>,<span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">y</font></span><a href="complfld.html#K3">.|</a></span>,<span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


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