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<b>let </b><font color="Maroon" title="c1">a1</font> be    <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> ;<br/><b>let </b><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">a2</font> be    <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/><b>let </b><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="E1:72"/><i><font color="Green" title="E62">A1</font></i>: 
( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a2</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="c2">y1</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="E2:72"/>
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="c2">y1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="complsp1.html#K1" title="COMPLSP1:func.1">multcomplex</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a2</font> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <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="72_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:72_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="c2">y1</font>
 ;<br/>

<i><font color="Green" title="E63">A2</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c2">y1</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">i</font> = 
<font color="Maroon" title="c1">a1</font> <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>
<b>by </b><i><a class="ref" href="complsp2.html#T16" target="_self" title="COMPLSP2:th.16">Th16</a></i>
<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> <font color="Maroon" title="c1">a1</font>,<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/>.= 
<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> <font color="Maroon" title="c1">a1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<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>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">[;]</a> <font color="Maroon" title="c1">a1</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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>
<b>by </b><i><a class="ref" href="funcop_1.html#T66" title="FUNCOP_1:th.66">FUNCOP_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="complsp1.html#K1" title="COMPLSP1:func.1">multcomplex</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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>
<b>by </b><i><a class="ref" href="complsp1.html#D5" title="COMPLSP1:def.5">COMPLSP1:def 5</a></i>
;<br/>
<i><font color="Green" title="E64">A3</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">a2</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c4">y2</font></span>)</span> <a href="complsp2.html#K10" title="COMPLSP2:func.10">.</a> <font color="Maroon" title="c5">i</font> = 
<font color="Maroon" title="c3">a2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <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="ref" href="rvsum_1.html#T66" title="RVSUM_1:th.66">RVSUM_1:66</a></i>
<br/>.= 
<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> <font color="Maroon" title="c3">a2</font>,<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="ref" href="binop_2.html#D11" title="BINOP_2:def.11">BINOP_2:def 11</a></i>
<br/>.= 
<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> <font color="Maroon" title="c3">a2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<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></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">[;]</a> <font color="Maroon" title="c3">a2</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <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="ref" href="funcop_1.html#T66" title="FUNCOP_1:th.66">FUNCOP_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a2</font> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <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="ref" href="rvsum_1.html#D3" title="RVSUM_1:def.3">RVSUM_1:def 3</a></i>
;<br/>
<a NAME="E4:72_1"/>
<font color="Maroon" title="c1">a1</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c2">y1</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="ref" href="complsp2.html#T16" target="_self" title="COMPLSP2:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E5:72_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="complsp1.html#K1" title="COMPLSP1:func.1">multcomplex</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a2</font> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <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="complsp2.html#E1:72"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="complsp2.html#E2:72_1"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="complsp2.html#E3:72_1"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="rvsum_1.html#T66" title="RVSUM_1:th.66">RVSUM_1:66</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:72"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="complsp1.html#K1" title="COMPLSP1:func.1">multcomplex</a> </span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a2</font> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</a> </span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c4">y2</font>
 <b>by </b><i><a class="txt" href="complsp2.html#E1:72"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="complsp2.html#T55" target="_self" title="COMPLSP2:th.55">Th55</a></i>;<br/>


</div>
