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

<span class="kw">let </span><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">y1</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x2</font>, <font color="Olive" title="b2">y2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x2</font> &amp; <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y2</font> &amp;  <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="Olive" title="b2">y2</font> holds <br/><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">y2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c4">y2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> &amp; <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp;  <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="c4">y2</font> implies <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y2</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font>
 <span class="kw">and </span><br/><a NAME="E2:46"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font>
 <span class="kw">and </span><br/><a NAME="E3:46"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
 <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="c4">y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y2</font>)</span><br/>

<a NAME="E4:46"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">x1</font> holds <br/><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">y1</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">x1</font> implies <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<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="c5">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">y1</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>) )</span><br/>

<a NAME="E1:46_1"/>
<span class="p1">(<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="c5">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y1</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<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="c5">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">y1</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">i</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D5" title="BINOP_2:def.5">BINOP_2:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:46_1"/>
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">x1</font> implies <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<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="c5">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">y1</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="binop_2.html#D11" title="BINOP_2:def.11">BINOP_2:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:46"/>
<a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="complsp2.html#T45" title="COMPLSP2:th.45">COMPLSP2:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
