<?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="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></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="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span></span><br/>





<a NAME="E1:71"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</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="E2:71"/>
 <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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></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="ref" href="csspace.html#D13" target="_self" onmouseover="rs('csspace/D13')" onmouseout="rh()">Def11</a></span>;<br/>
<a NAME="E3:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">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"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</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>  <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#T34" target="_self" onmouseover="rs('csspace/T34')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E4:71"/>
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></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="csspace.html#T44" target="_self" onmouseover="rs('csspace/T44')" onmouseout="rh()">Th39</a>, <a class="ref" href="square_1.html#T22" onmouseover="rs('square_1/T22')" onmouseout="rh()">SQUARE_1:22</a></span>;<br/>
<a NAME="E5:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <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"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></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="#E3:71"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="square_1.html#T28" onmouseover="rs('square_1/T28')" onmouseout="rh()">SQUARE_1:28</a></span>;<br/>
<a NAME="E6:71"/>
<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="E7:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A4</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> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D2" onmouseover="rs('square_1/D2')" onmouseout="rh()">SQUARE_1:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E50">A5</font></span>:  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <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="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="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T27" onmouseover="rs('complex1/T27')" onmouseout="rh()">COMPLEX1:27</a></span>
<br/>.= 
 <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="c5">x</font></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 href="complex1.html#K4" title="COMPLEX1:func.4">Im</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="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K2" title="XCMPLX_0:func.2">+</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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><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="c5">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K2" title="XCMPLX_0:func.2">+</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="real_1.html#K3" title="REAL_1:func.3">+</a> <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="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="ref" href="complex1.html#T8" onmouseover="rs('complex1/T8')" onmouseout="rh()">COMPLEX1:8</a></span>
<br/>.= 
<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"><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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="real_1.html#K3" title="REAL_1: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"><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="real_1.html#K3" title="REAL_1:func.3">+</a> <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="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="ref" href="complex1.html#T8" onmouseover="rs('complex1/T8')" onmouseout="rh()">COMPLEX1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</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="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</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="real_1.html#K3" title="REAL_1: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"><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="real_1.html#K3" title="REAL_1:func.3">+</a> <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="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="ref" href="complex1.html#T8" onmouseover="rs('complex1/T8')" onmouseout="rh()">COMPLEX1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</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"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</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"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</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="ref" href="csspace.html#D13" target="_self" onmouseover="rs('csspace/D13')" onmouseout="rh()">Def11</a></span>
<br/>.= 
<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="real_1.html#K3" title="REAL_1: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"><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> <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="E10:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<span class="p1">(<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="c5">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K2" title="XCMPLX_0:func.2">+</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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> <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"><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="c5">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K2" title="XCMPLX_0:func.2">+</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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c6">y</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"><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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="complex1.html#T13" onmouseover="rs('complex1/T13')" onmouseout="rh()">COMPLEX1:13</a></span>;<br/>
<span class="lab"><font color="Green" title="E52">A7</font></span>:  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</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> = 
 <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="complex1.html#K8" title="COMPLEX1:func.8">*'</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T27" onmouseover="rs('complex1/T27')" onmouseout="rh()">COMPLEX1:27</a></span>
<br/>.= 
 <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="c5">x</font></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="E12:71"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</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> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#T54" onmouseover="rs('complex1/T54')" onmouseout="rh()">COMPLEX1:54</a></span>;<br/>
<a NAME="E13:71"/>
<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"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T35" target="_self" onmouseover="rs('csspace/T35')" onmouseout="rh()">Th30</a></span>;<br/>
<a NAME="E14:71"/><span class="kw">then </span>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</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> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:71"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E15:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A9</font></span>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0: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"><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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E16:71"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</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="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K2" title="XCMPLX_0:func.2">+</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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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> <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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T31" target="_self" onmouseover="rs('csspace/T31')" onmouseout="rh()">Th26</a></span>;<br/>
<a NAME="E17:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A10</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"><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="c5">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K2" title="XCMPLX_0:func.2">+</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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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> <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 href="complex1.html#K3" title="COMPLEX1:func.3">Re</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="c5">x</font> <a href="csspace.html#K10" title="CSSPACE:func.10">.|.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K2" title="XCMPLX_0:func.2">+</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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="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"><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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K2" title="XCMPLX_0:func.2">+</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="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"><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="ref" href="complex1.html#T8" onmouseover="rs('complex1/T8')" onmouseout="rh()">COMPLEX1:8</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> <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="c5">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="real_1.html#K3" title="REAL_1:func.3">+</a> <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="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="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"><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="ref" href="complex1.html#T8" onmouseover="rs('complex1/T8')" onmouseout="rh()">COMPLEX1:8</a></span>
<br/>.= 
<span class="p1">(<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="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="real_1.html#K3" title="REAL_1:func.3">+</a> <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="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="real_1.html#K3" title="REAL_1:func.3">+</a> <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="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="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"><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="ref" href="complex1.html#T8" onmouseover="rs('complex1/T8')" onmouseout="rh()">COMPLEX1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><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="c5">x</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="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="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"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</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"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</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="ref" href="csspace.html#T34" target="_self" onmouseover="rs('csspace/T34')" onmouseout="rh()">Th29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><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="c5">x</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="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="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"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</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"><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T34" target="_self" onmouseover="rs('csspace/T34')" onmouseout="rh()">Th29</a></span>
;<br/>
<a NAME="E19:71"/><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"><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="c5">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K2" title="XCMPLX_0:func.2">+</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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c6">y</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">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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="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="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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:71"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E17:71"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E10:71"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E20:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A11</font></span>: 
<span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K2" title="XCMPLX_0:func.2">+</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="#E5:71"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="csspace.html#T31" target="_self" onmouseover="rs('csspace/T31')" onmouseout="rh()">Th26</a></span>;<br/>
<a NAME="E21:71"/><span class="lab"><font color="Green" title="E57">A12</font></span>: 
<span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</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="csspace.html#T44" target="_self" onmouseover="rs('csspace/T44')" onmouseout="rh()">Th39</a></span>;<br/>
<a NAME="E22:71"/>
<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="E23:71"/><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="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <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"><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 class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D2" onmouseover="rs('square_1/D2')" onmouseout="rh()">SQUARE_1:def 2</a></span>;<br/>
<a NAME="E24:71"/><span class="kw">then </span>
<span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></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"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></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="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="#E20:71"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="#E15:71"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E25:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A13</font></span>: 
<span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:71"><span class="lab"><font color="Green" title="E49">A4</font></span></a></span>;<br/>
<a NAME="E26:71"/>
<span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</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="csspace.html#T44" target="_self" onmouseover="rs('csspace/T44')" onmouseout="rh()">Th39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:71"/>
<span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:71"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E25:71"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="ref" href="square_1.html#T16" onmouseover="rs('square_1/T16')" onmouseout="rh()">SQUARE_1:16</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>
