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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <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>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</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">y</font> holds <br/> <a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">a</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</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">y</font> implies  <a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E26">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</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">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">x'</font> = <font color="Maroon" title="c1">x</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">x</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">y'</font> = <font color="Maroon" title="c2">y</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">y</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/>
 <a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c2">y</font> = 
<font color="Maroon" title="c3">a</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c4">x'</font>,<font color="Maroon" title="c5">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:34"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="matrixc1.html#T16" target="_self" title="MATRIXC1:th.16">Th16</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span>

;<br/>
<b>hence </b><a NAME="E5:34"/>
 <a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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