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<span class="kw">let </span><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</font>, <font color="Maroon" title="c3">y1</font>, <font color="Maroon" title="c4">y2</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="c1">x1</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">x2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x2</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">y1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y1</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="c4">y2</font> implies <span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">y1</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c4">y2</font></span>)</span></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c3">y1</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p3"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c4">y2</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p2"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">y1</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</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">x2</font>
 <span class="kw">and </span><br/><a NAME="E2:70"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x2</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">y1</font>
 <span class="kw">and </span><br/><a NAME="E3:70"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y1</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="c4">y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">y1</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c4">y2</font></span>)</span></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c3">y1</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p3"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c4">y2</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p2"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">y1</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span></span><br/>

<a NAME="E4:70"/>
<span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">x2</font></span>)</span>,<font color="Maroon" title="c4">y2</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c4">y2</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="complsp2.html#T66" target="_self" title="COMPLSP2:th.66">Th66</a></span>;<br/>
<a NAME="E5:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A4</font></span>: 
<span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">x2</font></span>)</span>,<font color="Maroon" title="c3">y1</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">x2</font></span>)</span>,<font color="Maroon" title="c4">y2</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c3">y1</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p2"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">y1</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c4">y2</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p2"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="complsp2.html#T66" target="_self" title="COMPLSP2:th.66">Th66</a></span>;<br/>
<a NAME="E6:70"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">x2</font></span>)</span> <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="c1">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="complsp2.html#T6" target="_self" title="COMPLSP2:th.6">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:70"/>
<span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">y1</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c4">y2</font></span>)</span></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c3">y1</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p3"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c4">y2</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p2"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">y1</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E5:70"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="complsp2.html#T71" target="_self" title="COMPLSP2:th.71">Th71</a></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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