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

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E4:4"/><i><font color="Green">E23</font></i>: 
 <a href="polyeq_2.html#K1" target="_self">Polynom</a> <font color="Maroon">a</font>,0,<font color="Maroon">c</font>,0,<font color="Maroon">e</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E5:4"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="newton.html#K2">|^</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="newton.html#K2">|^</a> 3</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:4_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">0 <a href="newton.html#K3">|^</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">0 <a href="newton.html#K3">|^</a> 3</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">0 <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:4_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">0 <a href="newton.html#K3">|^</a> 3</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><b>hence </b><a NAME="E4:4_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1"><b>suppose </b></a><a NAME="E1:4_2_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><font color="Maroon">x</font> <a href="newton.html#K2">|^</a> 4 = 
<font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K2">+</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T47">POWER:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>
;<br/><a NAME="E3:4_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E4:4_2_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="polyeq_1.html#K3">Polynom</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="polyeq_1.html#D2">POLYEQ_1:def 2</a></i>;<br/><a NAME="E5:4_2_1"/>
 <a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="quin_1.html#D1">QUIN_1:def 1</a></i>;<br/><a NAME="E6:4_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="square_1.html#K3">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> or <font color="Maroon">x</font> <a href="square_1.html#K3">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="polyeq_1.html#T5">POLYEQ_1:5</a></i>;<br/><a NAME="E7:4_2_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> or  <a href="complex1.html#K18">abs</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>;<br/><b>hence </b><a NAME="E8:4_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_2"><b>suppose </b></a><a NAME="E1:4_2_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:4_2_2"/><b>then </b><i><font color="Green">E30</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:4_2_2"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="newton.html#K2">|^</a> 4 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="newton.html#K2">|^</a> 4
 <b>by </b><i>, <a class="ref" href="power.html#T1">POWER:1</a></i>;<br/><b>then </b><font color="Maroon">x</font> <a href="newton.html#K2">|^</a> 4 = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K2">+</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T47">POWER:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="power.html#K3">to_power</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="power.html#K3">to_power</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="power.html#K3">to_power</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>
;<br/><a NAME="E5:4_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E6:4_2_2"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="polyeq_1.html#K3">Polynom</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="polyeq_1.html#D2">POLYEQ_1:def 2</a></i>;<br/><a NAME="E7:4_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="quin_1.html#D1">QUIN_1:def 1</a></i>;<br/><a NAME="E8:4_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="square_1.html#K3">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> or <font color="Maroon">x</font> <a href="square_1.html#K3">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="polyeq_1.html#T5">POLYEQ_1:5</a></i>;<br/><a NAME="E9:4_2_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> or  <a href="complex1.html#K18">abs</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>;<br/><a NAME="E10:4_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> or <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E11:4_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
