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<b>let </b><font color="Maroon" title="c1">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</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>  holds  <span class="p1">(<span class="default"><a href="complsp1.html#K5" title="COMPLSP1:func.5">-</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">x</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"> <span class="p1">(<span class="default"><a href="complsp1.html#K5" title="COMPLSP1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font></span>)</span></span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1"/>
(  not <font color="Maroon" title="c1">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> <span class="p1">(<span class="default"><a href="complsp1.html#K5" title="COMPLSP1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span> or <font color="Maroon" title="c1">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> <span class="p1">(<span class="default"><a href="complsp1.html#K5" title="COMPLSP1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><b>suppose </b></a><a NAME="E1:12_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
not <font color="Maroon" title="c1">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> <span class="p1">(<span class="default"><a href="complsp1.html#K5" title="COMPLSP1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="complsp1.html#K5" title="COMPLSP1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font></span>)</span></span><br/><div class="add"><a NAME="E2:12_1_1"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
not <font color="Maroon" title="c1">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">x</font>
 <b>by </b><i><a class="ref" href="complsp2.html#T4" target="_self" title="COMPLSP2:th.4">Th4</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="complsp1.html#K5" title="COMPLSP1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font> = 
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E1:12_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>
<br/>.= 
 <a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:12_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2"><b>suppose </b></a><a NAME="E1:12_1_2"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c1">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> <span class="p1">(<span class="default"><a href="complsp1.html#K5" title="COMPLSP1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="complsp1.html#K5" title="COMPLSP1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font></span>)</span></span><br/><div class="add"><b>set </b><font color="Maroon" title="c3">r</font> = <font color="Maroon" title="c2">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font>;<br/><a NAME="E2:12_1_2"/>
 <a href="complsp1.html#K5" title="COMPLSP1:func.5">-</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K25" title="BINOP_2:func.25">compcomplex</a>  <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="complsp1.html#D9" title="COMPLSP1:def.9">COMPLSP1:def 9</a></i>;<br/><a NAME="E3:12_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complsp1.html#K5" title="COMPLSP1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K25" title="BINOP_2:func.25">compcomplex</a>  <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:12_1_2"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><b>hence </b><a NAME="E4:12_1_2"/>
<span class="p1">(<span class="default"><a href="complsp1.html#K5" title="COMPLSP1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="binop_2.html#D1" title="BINOP_2:def.1">BINOP_2:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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