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<div class="add">

<b>let </b><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</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="E1:24"/><i><font color="Green" title="E21">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</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">x2</font>
 ;<br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green" title="E22">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Maroon" title="c2">x2</font></span>)</span> <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="c1">x1</font>
 
<b>by </b><i><a class="ref" href="complsp2.html#T6" target="_self" title="COMPLSP2:th.6">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">x'</font> = <font color="Maroon" title="c1">x1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">y'</font> = <font color="Maroon" title="c2">x2</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x3</font> = <font color="Maroon" title="c1">x1</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a>  as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x4</font> = <font color="Maroon" title="c2">x2</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a>  as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E7:24"/><i><font color="Green" title="E23">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></i>;<br/>
<a NAME="E8:24"/><i><font color="Green" title="E24">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></i>;<br/>
<a NAME="E9:24"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <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="c1">x1</font>
 
<b>by </b><i><a class="txt" href="complsp2.html#E1:24"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="complsp2.html#E7:24"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="complsp2.html#T6" target="_self" title="COMPLSP2:th.6">Th6</a></i>;<br/>
<a NAME="E10:24"/><b>then </b><i><font color="Green" title="E25">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="complsp2.html#E2:24"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:24_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E2:24_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> )
 ;<br/><a NAME="E3:24_1"/><b>then </b><i><font color="Green" title="E27">A7</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Maroon" title="c2">x2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x'</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">y'</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="txt" href="complsp2.html#E1:24"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="complsp2.html#E1:24_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="complsp2.html#T18" target="_self" title="COMPLSP2:th.18">Th18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</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"><font color="Maroon" title="c2">x2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></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="c1">x1</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</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"><font color="Maroon" title="c2">x2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span>
<b>by </b><i><a class="txt" href="complsp2.html#E2:24"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="complsp2.html#E3:24_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</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"><font color="Maroon" title="c2">x2</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="txt" href="complsp2.html#E1:24"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="complsp2.html#E2:24"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="complsp2.html#E3:24_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x3</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Maroon" title="c6">x4</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font>
<b>by </b><i><a class="txt" href="complsp2.html#E1:24"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="complsp2.html#E7:24"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="complsp2.html#E8:24"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="complsp2.html#E1:24_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="complsp2.html#T18" target="_self" title="COMPLSP2:th.18">Th18</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:24"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>
 <b>by </b><i><a class="txt" href="complsp2.html#E10:24"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div>
