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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:86"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>, <font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>, <font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>, <font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:86"/><i><font color="Green">E75</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#D3">COMPLEX1:def 3</a></i>;<br/>
<a NAME="E4:86"/><i><font color="Green">E76</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#D3">COMPLEX1:def 3</a></i>;<br/>
<a NAME="E5:86"/><i><font color="Green">E77</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#D3">COMPLEX1:def 3</a></i>;<br/>
<a NAME="E6:86"/><i><font color="Green">E78</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#D3">COMPLEX1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>, <font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>, <font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>, <font color="Maroon">c<sub>14</sub></font> = <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/>
<span class="p1"><a href="complsp2.html#K8" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="complsp2.html#K8" target="_self">)|</a></span> <a href="complex1.html#K15">*'</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p0">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span> <a href="complex1.html#K15">*'</a> 

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K3">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="binop_2.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_2.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K3">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="binop_2.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_2.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T118">COMPLEX1:118</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="binop_2.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="binop_2.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_2.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T118">COMPLEX1:118</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="binop_2.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_2.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T124">COMPLEX1:124</a>, <a class="txt" href="complsp2.html#E3:86"><i><font color="Green">E75</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K3">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="binop_2.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_2.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T124">COMPLEX1:124</a>, <a class="txt" href="complsp2.html#E6:86"><i><font color="Green">E78</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K3">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K1">-</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_2.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T116">COMPLEX1:116</a>, <a class="ref" href="complex1.html#T121">COMPLEX1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K3">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K1">-</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="binop_2.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T120">COMPLEX1:120</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K3">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K1">-</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_2.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T124">COMPLEX1:124</a>, <a class="txt" href="complsp2.html#E5:86"><i><font color="Green">E77</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K3">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K1">-</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_2.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T124">COMPLEX1:124</a>, <a class="txt" href="complsp2.html#E4:86"><i><font color="Green">E76</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>

<br/>.= 
<span class="p1"><a href="complsp2.html#K8" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="complsp2.html#K8" target="_self">)|</a></span>

;<br/>
<b>hence </b><a NAME="E9:86"/>
<span class="p1"><a href="complsp2.html#K8" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="complsp2.html#K8" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complsp2.html#K8" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="complsp2.html#K8" target="_self">)|</a></span> <a href="complex1.html#K15">*'</a> 
 ;<br/>


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