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<span class="kw">let </span><font color="Maroon" title="c1">z</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">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">v</font>, <font color="Olive" title="b2">w</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a>,<a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a>,<a href="csspace.html#K4" title="CSSPACE:func.4">l_add</a>,<a href="csspace.html#K5" title="CSSPACE:func.5">l_mult</a> #) holds  <font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">w</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c3">w</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a>,<a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a>,<a href="csspace.html#K4" title="CSSPACE:func.4">l_add</a>,<a href="csspace.html#K5" title="CSSPACE:func.5">l_mult</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="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">w</font></span>)</span></span><br/>



<span class="lab"><font color="Green" title="E13">A1</font></span>: <font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">w</font></span>)</span> = 
<font color="Maroon" title="c1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T3" target="_self" title="CSSPACE:th.3">Th5</a></span>
<br/>.= 
<font color="Maroon" title="c1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D4" target="_self" title="CSSPACE:def.4">Def4</a></span>
<br/>.= 
<font color="Maroon" title="c1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T1" target="_self" title="CSSPACE:th.1">Th3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_1.html#T16" title="COMSEQ_1:th.16">COMSEQ_1:16</a></span>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">w</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D4" target="_self" title="CSSPACE:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span></span>)</span> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T3" target="_self" title="CSSPACE:th.3">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span></span>)</span> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T3" target="_self" title="CSSPACE:th.3">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T1" target="_self" title="CSSPACE:th.1">Th3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T1" target="_self" title="CSSPACE:th.1">Th3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:13"/>
<font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E13">A1</font></span></a></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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