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

<span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="csspace.html#NM1" title="CSSPACE:NM.1">ComplexUnitarySpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X</font> holds  <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><br/>





<a NAME="E1:60"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">X</font>) implies <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:60_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">X</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><br/>

<a NAME="E2:60_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D13" target="_self" onmouseover="rs('csspace/D13')" onmouseout="rh()">Def11</a></span>;<br/>
<a NAME="E3:60_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T45" onmouseover="rs('complex1/T45')" onmouseout="rh()">COMPLEX1:45</a></span>;<br/>
<a NAME="E4:60_1"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" onmouseover="rs('xreal_1/T63')" onmouseout="rh()">XREAL_1:63</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">c2</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">c1</font> = <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D2" onmouseover="rs('xcmplx_0/D2')" onmouseout="rh()">XCMPLX_0:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E44">A4</font></span>:  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c8">c1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c8">c1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T9" onmouseover="rs('complex1/T9')" onmouseout="rh()">COMPLEX1:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c8">c1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T12" onmouseover="rs('complex1/T12')" onmouseout="rh()">COMPLEX1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c8">c1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<span class="lab"><font color="Green" title="E45">A5</font></span>:  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T17" onmouseover="rs('complex1/T17')" onmouseout="rh()">COMPLEX1:17</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D13" target="_self" onmouseover="rs('csspace/D13')" onmouseout="rh()">Def11</a></span>
;<br/>
<a NAME="E8:60_1"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T40" onmouseover="rs('complex1/T40')" onmouseout="rh()">COMPLEX1:40</a></span>;<br/>
<a NAME="E9:60_1"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D13" target="_self" onmouseover="rs('csspace/D13')" onmouseout="rh()">Def11</a></span>;<br/>
<a NAME="E10:60_1"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D13" target="_self" onmouseover="rs('csspace/D13')" onmouseout="rh()">Def11</a></span>;<br/>
<a NAME="E11:60_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E12:60_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A9</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T13" onmouseover="rs('complex1/T13')" onmouseout="rh()">COMPLEX1:13</a></span>;<br/>
<a NAME="E13:60_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A10</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="complex1.html#T13" onmouseover="rs('complex1/T13')" onmouseout="rh()">COMPLEX1:13</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> = 
<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" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="csspace.html#E58"><span class="lab"><font color="Green" title="E38">Lm10</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E12:60_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="complex1.html#T13" onmouseover="rs('complex1/T13')" onmouseout="rh()">COMPLEX1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:60_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="csspace.html#D13" target="_self" onmouseover="rs('csspace/D13')" onmouseout="rh()">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:60_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="csspace.html#D13" target="_self" onmouseover="rs('csspace/D13')" onmouseout="rh()">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E15:60_1"/><span class="kw">then </span>
( (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c8">c1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) or (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c8">c1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="csspace.html#D13" target="_self" onmouseover="rs('csspace/D13')" onmouseout="rh()">Def11</a></span>;<br/>
<a NAME="E16:60_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="#E9:60_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E12:60_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E3:60_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="complex1.html#T8" onmouseover="rs('complex1/T8')" onmouseout="rh()">COMPLEX1:8</a>, <a class="ref" href="complex1.html#T13" onmouseover="rs('complex1/T13')" onmouseout="rh()">COMPLEX1:13</a></span>;<br/>
<a NAME="E17:60_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:60_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a></span>;<br/>
<a NAME="E18:60_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A11</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/>
<a NAME="E19:60_1"/><span class="lab"><font color="Green" title="E52">A12</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D13" target="_self" onmouseover="rs('csspace/D13')" onmouseout="rh()">Def11</a></span>;<br/>
<a NAME="E20:60_1"/><span class="lab"><font color="Green" title="E53">A13</font></span>: 
<span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" onmouseover="rs('xreal_1/T63')" onmouseout="rh()">XREAL_1:63</a></span>;<br/>
<a NAME="E21:60_1"/>
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c8">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E12:60_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="complex1.html#T13" onmouseover="rs('complex1/T13')" onmouseout="rh()">COMPLEX1:13</a></span>;<br/>
<a NAME="E22:60_1"/><span class="kw">then </span>
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c8">c1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:60_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="complex1.html#T9" onmouseover="rs('complex1/T9')" onmouseout="rh()">COMPLEX1:9</a></span>;<br/>
<a NAME="E23:60_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A14</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T50" onmouseover="rs('complex1/T50')" onmouseout="rh()">COMPLEX1:50</a></span>;<br/>
<a NAME="E24:60_1"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T40" onmouseover="rs('complex1/T40')" onmouseout="rh()">COMPLEX1:40</a></span>;<br/>
<a NAME="E25:60_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A15</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E20:60_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a></span>;<br/>
<a NAME="E26:60_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E27:60_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:60_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E25:60_1"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E28:60_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:60_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E23:60_1"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="ref" href="complex1.html#T50" onmouseover="rs('complex1/T50')" onmouseout="rh()">COMPLEX1:50</a></span>;<br/>
<a NAME="E29:60_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:60_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a></span>;<br/>
<a NAME="E30:60_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a></span>;<br/>
<a NAME="E31:60_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A16</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T53" onmouseover="rs('complex1/T53')" onmouseout="rh()">COMPLEX1:53</a></span>;<br/>
<a NAME="E32:60_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" onmouseover="rs('xreal_1/T63')" onmouseout="rh()">XREAL_1:63</a></span>;<br/>
<a NAME="E33:60_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A17</font></span>: 
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:60_1"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="ref" href="square_1.html#T26" onmouseover="rs('square_1/T26')" onmouseout="rh()">SQUARE_1:26</a></span>;<br/>
<a NAME="E34:60_1"/><span class="lab"><font color="Green" title="E58">A18</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a></span>;<br/>
<a NAME="E35:60_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a></span>;<br/>
<a NAME="E36:60_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:60_1"><span class="lab"><font color="Green" title="E57">A17</font></span></a>, <a class="txt" href="#E34:60_1"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="ref" href="square_1.html#T29" onmouseover="rs('square_1/T29')" onmouseout="rh()">SQUARE_1:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E37:60_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a>, <a class="ref" href="square_1.html#T22" onmouseover="rs('square_1/T22')" onmouseout="rh()">SQUARE_1:22</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:60"/>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">X</font>) implies <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:60_2"/><span class="lab"><font color="Green" title="E42">A19</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">X</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><br/>

<a NAME="E2:60_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D13" target="_self" onmouseover="rs('csspace/D13')" onmouseout="rh()">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:60_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_2"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="csspace.html#T30" target="_self" onmouseover="rs('csspace/T30')" onmouseout="rh()">Th25</a>, <a class="ref" href="complex1.html#T44" onmouseover="rs('complex1/T44')" onmouseout="rh()">COMPLEX1:44</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">hence </span><a NAME="E3:60"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E41">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
