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<b>let </b><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> ;<br/><b>let </b><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> ;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E35">A1</font></i>: 
( <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> )
 ;<br/>

<a NAME="E2:47"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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#K2" title="BINOP_1:func.2">.</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#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="complsp2.html#K10" title="COMPLSP2:func.10">.</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="complsp2.html#K10" title="COMPLSP2:func.10">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:47_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#K2" title="BINOP_1:func.2">.</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>
 
<b>by </b><i><a class="ref" href="binop_2.html#D5" title="BINOP_2:def.5">BINOP_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E2:47_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#K2" title="BINOP_1:func.2">.</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#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="complsp2.html#K10" title="COMPLSP2:func.10">.</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="complsp2.html#K10" title="COMPLSP2:func.10">.</a> <font color="Maroon" title="c5">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:47"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="binop_2.html#D11" title="BINOP_2:def.11">BINOP_2:def 11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:47"/>
<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>
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:47"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="complsp2.html#T45" title="COMPLSP2:th.45">COMPLSP2:45</a></i>;<br/>


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