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<b>let </b><font color="Maroon" title="c1">c</font> be  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">x</font>, <font color="Olive" title="b2">y</font> being   <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>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b2">y</font> holds <br/><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">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="c3">y</font> implies <font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E22">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">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="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">x'</font> = <font color="Maroon" title="c2">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="c2">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="c3">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="c3">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/>
<b>set </b><font color="Maroon" title="c6">cM</font> = <font color="Maroon" title="c1">c</font> <a href="complsp1.html#K1" title="COMPLSP1:func.1">multcomplex</a> ;<br/>
<a NAME="E4:31"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c1">c</font> is    <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> 
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></i>;<br/>
<font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Maroon" title="c3">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K1" title="COMPLSP1:func.1">multcomplex</a> </span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Maroon" title="c3">y</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp1.html#D10" title="COMPLSP1:def.10">COMPLSP1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K1" title="COMPLSP1:func.1">multcomplex</a> </span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp1.html#D7" title="COMPLSP1:def.7">COMPLSP1:def 7</a></i>
<br/>.= 
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">c</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="c4">x'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">c</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="c5">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="complsp2.html#E1:31"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="complsp2.html#E4:31"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="complsp1.html#T17" title="COMPLSP1:th.17">COMPLSP1:17</a>, <a class="ref" href="finseqop.html#T52" title="FINSEQOP:th.52">FINSEQOP:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">c</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">x</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">c</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="c3">y</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp1.html#D7" title="COMPLSP1:def.7">COMPLSP1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">c</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="c3">y</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp1.html#D10" title="COMPLSP1:def.10">COMPLSP1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp1.html#D10" title="COMPLSP1:def.10">COMPLSP1:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E6:31"/>
<font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <font color="Maroon" title="c3">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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