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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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">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="c2">y</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</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">z</font> implies <font color="Maroon" title="c1">x</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c3">z</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c5">y1</font> = <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c6">z1</font> = <font color="Maroon" title="c3">z</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="complsp2.html#E8"><span class="lab"><font color="Green" title="E5">Lm4</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:30"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">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="c2">y</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</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">z</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">x</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c3">z</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">z9</font> = <font color="Maroon" title="c6">z1</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">z</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="c8">y9</font> = <font color="Maroon" title="c5">y1</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">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="c9">x9</font> = <font color="Maroon" title="c4">x1</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="c1">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/>
<font color="Maroon" title="c1">x</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span> = 
<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="c4">x1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c6">z1</font></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> <font color="Maroon" title="c4">x1</font>,<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="c5">y1</font>,<font color="Maroon" title="c6">z1</font></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"><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="c9">x9</font>,<font color="Maroon" title="c8">y9</font></span>)</span>,<font color="Maroon" title="c7">z9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="finseqop.html#T29" title="FINSEQOP:th.29">FINSEQOP:29</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"><font color="Maroon" title="c4">x1</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<font color="Maroon" title="c6">z1</font>
<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">x</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c3">z</font>
<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="kw">hence </span><a NAME="E7:30"/>
<font color="Maroon" title="c1">x</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c3">z</font>
 ; <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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