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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</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="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Olive" title="b2">y</font></span>)</span></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</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="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c3">y</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
 <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</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="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c3">y</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">cc</font> = <font color="Maroon" title="c1">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/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">y9</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">x9</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">cM</font> = <font color="Maroon" title="c4">cc</font> <a href="seq_4.html#K7" title="SEQ_4:func.7">multcomplex</a> ;<br/>
<font color="Maroon" title="c1">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">cc</font> <a href="seq_4.html#K7" title="SEQ_4:func.7">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="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p2">(<span class="default"><a href="seq_4.html#K11" title="SEQ_4:func.11">-</a> <font color="Maroon" title="c3">y</font></span>)</span></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"><font color="Maroon" title="c4">cc</font> <a href="seq_4.html#K7" title="SEQ_4:func.7">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>,<span class="p2">(<span class="default"><a href="seq_4.html#K11" title="SEQ_4:func.11">-</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D9" title="SEQ_4:def.9">SEQ_4:def 9</a></span>
<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="c4">cc</font> <a href="seq_4.html#K7" title="SEQ_4:func.7">multcomplex</a> </span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">x9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">cc</font> <a href="seq_4.html#K7" title="SEQ_4:func.7">multcomplex</a> </span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p2">(<span class="default"><a href="complsp2.html#K5" title="COMPLSP2:func.5">-</a> <font color="Maroon" title="c5">y9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="seq_4.html#T73" title="SEQ_4:th.73">SEQ_4:73</a>, <a class="ref" href="finseqop.html#T52" title="FINSEQOP:th.52">FINSEQOP:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">cc</font> <a href="seq_4.html#K7" title="SEQ_4:func.7">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="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">cc</font> <a href="seq_4.html#K7" title="SEQ_4:func.7">multcomplex</a> </span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p2">(<span class="default"><a href="seq_4.html#K11" title="SEQ_4:func.11">-</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D9" title="SEQ_4:def.9">SEQ_4:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">cc</font> <a href="seq_4.html#K7" title="SEQ_4:func.7">multcomplex</a> </span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p2">(<span class="default"><a href="seq_4.html#K11" title="SEQ_4:func.11">-</a> <font color="Maroon" title="c3">y</font></span>)</span></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"><font color="Maroon" title="c1">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p2">(<span class="default"><a href="seq_4.html#K11" title="SEQ_4:func.11">-</a> <font color="Maroon" title="c3">y</font></span>)</span></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"><font color="Maroon" title="c1">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c3">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#T42" target="_self" title="COMPLSP2:th.42">Th42</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:43"/>
<font color="Maroon" title="c1">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</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="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c3">y</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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