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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font> implies  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">z</font> <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="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">z</font> <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="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font>, <font color="Maroon" title="c6">y1</font>, <font color="Maroon" title="c7">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:23"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E4:23"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E5:23"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5" title="XCMPLX_0:def.5">XCMPLX_0:def 5</a></i>;<br/>
<a NAME="E6:23"/><i><font color="Green" title="E27">A5</font></i>: 
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font> )
 
<b>by </b><i><a class="txt" href="complex1.html#E3:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="complex1.html#E4:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="complex1.html#E11"><i><font color="Green" title="E7">Lm3</font></i></a></i>;<br/>
<a NAME="E7:23"/><i><font color="Green" title="E28">A6</font></i>: 
(  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y2</font> )
 
<b>by </b><i><a class="txt" href="complex1.html#E3:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="complex1.html#E4:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="complex1.html#E11"><i><font color="Green" title="E7">Lm3</font></i></a></i>;<br/>
<b>thus </b> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">z</font> = 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex1.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="complex1.html#E5:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="complex1.html#E11"><i><font color="Green" title="E7">Lm3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex1.html#E33"><i><font color="Green" title="E17">Lm8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex1.html#E35"><i><font color="Green" title="E19">Lm10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex1.html#E34"><i><font color="Green" title="E18">Lm9</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font></span>)</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="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex1.html#E6:23"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="complex1.html#E7:23"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="complex1.html#E35"><i><font color="Green" title="E19">Lm10</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
