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<span class="kw">let </span><font color="Maroon" title="c1">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c</font> being   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Olive" title="b1">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K17" title="SEQ_4:func.17">0c</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K17" title="SEQ_4:func.17">0c</a> <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></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K17" title="SEQ_4:func.17">0c</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K17" title="SEQ_4:func.17">0c</a> <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></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">cc</font> = <font color="Maroon" title="c2">c</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></span>;<br/>
<a NAME="E2:41"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="seq_4.html#K17" title="SEQ_4:func.17">0c</a> <span class="p2">(<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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<font color="Maroon" title="c2">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K17" title="SEQ_4:func.17">0c</a> <span class="p2">(<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></span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Maroon" title="c3">cc</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="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K17" title="SEQ_4:func.17">0c</a> <span class="p2">(<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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="complsp2.html#E2"><span class="lab"><font color="Green" title="E2">Lm1</font></span></a></span>
<br/>.= 
<a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> (<font color="Maroon" title="c3">cc</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="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p2">(<span class="default"><a href="seq_4.html#K17" title="SEQ_4:func.17">0c</a> <span class="p3">(<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></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T34" title="FUNCOP_1:th.34">FUNCOP_1:34</a></span>
<br/>.= 
<a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> (<font color="Maroon" title="c3">cc</font>,<span class="p1">(<span class="default"><a href="seq_4.html#K17" title="SEQ_4:func.17">0c</a> <span class="p2">(<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></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="relat_1.html#T53" title="RELAT_1:th.53">RELAT_1:53</a></span>
<br/>.= 
<a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> (<font color="Maroon" title="c3">cc</font>,<span class="p1">(<span class="default"><span class="p2">(<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="complsp2.html#K9" title="COMPLSP2:func.9">|-&gt;</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D12" title="SEQ_4:def.12">SEQ_4:def 12</a></span>
<br/>.= 
<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="complsp2.html#K9" title="COMPLSP2:func.9">|-&gt;</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">cc</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseqop.html#T18" title="FINSEQOP:th.18">FINSEQOP:18</a></span>
<br/>.= 
<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="complsp2.html#K9" title="COMPLSP2:func.9">|-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">cc</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D5" title="BINOP_2:def.5">BINOP_2:def 5</a></span>
<br/>.= 
 <a href="seq_4.html#K17" title="SEQ_4:func.17">0c</a> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D12" title="SEQ_4:def.12">SEQ_4:def 12</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:41"/>
<font color="Maroon" title="c2">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K17" title="SEQ_4:func.17">0c</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K17" title="SEQ_4:func.17">0c</a> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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