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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:72"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">x1</font> = <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>;<br/>
<b>set </b><font color="Maroon" title="c4">x2</font> = <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>;<br/>
<b>set </b><font color="Maroon" title="c5">y1</font> = <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>;<br/>
<b>set </b><font color="Maroon" title="c6">y2</font> = <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x1'</font> = <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>, <font color="Maroon" title="c8">x2'</font> = <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>, <font color="Maroon" title="c9">y1'</font> = <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>, <font color="Maroon" title="c10">y2'</font> = <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> as    <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>  ;<br/>
<a NAME="E3:72"/><i><font color="Green" title="E56">A1</font></i>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c7">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="complex1.html#D3" title="COMPLEX1:def.3">COMPLEX1:def 3</a></i>;<br/>
<a NAME="E4:72"/><i><font color="Green" title="E57">A2</font></i>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c8">x2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="complex1.html#D3" title="COMPLEX1:def.3">COMPLEX1:def 3</a></i>;<br/>
<a NAME="E5:72"/><i><font color="Green" title="E58">A3</font></i>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c9">y1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="complex1.html#D3" title="COMPLEX1:def.3">COMPLEX1:def 3</a></i>;<br/>
<a NAME="E6:72"/><i><font color="Green" title="E59">A4</font></i>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c10">y2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="complex1.html#D3" title="COMPLEX1:def.3">COMPLEX1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x1'</font> = <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>, <font color="Maroon" title="c12">x2'</font> = <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>, <font color="Maroon" title="c13">y1'</font> = <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>, <font color="Maroon" title="c14">y2'</font> = <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></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>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></i>;<br/>
<span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">x1'</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c14">y2'</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">y1'</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Maroon" title="c12">x2'</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">x1'</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c14">y2'</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">y1'</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Maroon" title="c12">x2'</font></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T118" title="COMPLEX1:th.118">COMPLEX1:118</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">x1'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">y2'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">y1'</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Maroon" title="c12">x2'</font></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T118" title="COMPLEX1:th.118">COMPLEX1:118</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">x1'</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">y2'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">y1'</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Maroon" title="c12">x2'</font></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span>
<b>by </b><i><a class="txt" href="complsp2.html#E3:72"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="complex1.html#T124" title="COMPLEX1:th.124">COMPLEX1:124</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">x1'</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c14">y2'</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">y1'</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Maroon" title="c12">x2'</font></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span>
<b>by </b><i><a class="txt" href="complsp2.html#E6:72"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="complex1.html#T124" title="COMPLEX1:th.124">COMPLEX1:124</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">x1'</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c14">y2'</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">y1'</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Maroon" title="c12">x2'</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T116" title="COMPLEX1:th.116">COMPLEX1:116</a>, <a class="ref" href="complex1.html#T121" title="COMPLEX1:th.121">COMPLEX1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">x1'</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c14">y2'</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">y1'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">x2'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T120" title="COMPLEX1:th.120">COMPLEX1:120</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">x1'</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c14">y2'</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">y1'</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">x2'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complsp2.html#E5:72"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="complex1.html#T124" title="COMPLEX1:th.124">COMPLEX1:124</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">x1'</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c14">y2'</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">y1'</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Maroon" title="c12">x2'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complsp2.html#E4:72"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="complex1.html#T124" title="COMPLEX1:th.124">COMPLEX1:124</a></i>
<br/>.= 
<span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span>

;<br/>
<b>hence </b><a NAME="E9:72"/>
<span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
 ;<br/>


</div>
