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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>   <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>   <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>   <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>   <a href="clvect_1.html#L1" title="CLVECT_1:struct.1">CLSStruct</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V1</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b2">d1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">V1</font><br/>  for <font color="Olive" title="b3">A</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b1">V1</font><br/>  for <font color="Olive" title="b4">M</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<font color="Olive" title="b1">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Olive" title="b1">V1</font>  st <font color="Olive" title="b2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">V1</font> &amp; <font color="Olive" title="b4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<font color="Olive" title="b1">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/>(  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">M</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">M</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">M</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">M</font> #) is  <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">M</font> #) is  <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">M</font> #) is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V1</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">d1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">V1</font><br/>  for <font color="Olive" title="b2">A</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">V1</font><br/>  for <font color="Olive" title="b3">M</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<font color="Maroon" title="c2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">V1</font>  st <font color="Olive" title="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Olive" title="b3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<font color="Maroon" title="c2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/>(  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">M</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">M</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">M</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">M</font> #) is  <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">M</font> #) is  <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">M</font> #) is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">d1</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">V1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">V1</font><br/>  for <font color="Olive" title="b2">M</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<font color="Maroon" title="c2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">V1</font>  st <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<font color="Maroon" title="c2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/>(  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">M</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">M</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">M</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">M</font> #) is  <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">M</font> #) is  <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">M</font> #) is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">V1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<font color="Maroon" title="c2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">V1</font>  st <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<font color="Maroon" title="c2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/>(  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Olive" title="b1">M</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Olive" title="b1">M</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Olive" title="b1">M</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Olive" title="b1">M</font> #) is  <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Olive" title="b1">M</font> #) is  <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Olive" title="b1">M</font> #) is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Olive" title="b1">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">M</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<font color="Maroon" title="c2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">V1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<font color="Maroon" title="c2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies (  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  ) )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">D</font> = <font color="Maroon" title="c2">V1</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 <span class="kw">and </span><br/><a NAME="E2:7"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">V1</font>
 <span class="kw">and </span><br/><a NAME="E3:7"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<font color="Maroon" title="c2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">W</font> =  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #);<br/>
<div><span class="lab"><font color="Green" title="E6">A4</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a><br/>  for <font color="Olive" title="b2">x</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) holds  <font color="Olive" title="b1">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">x</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">a</font> be   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) holds  <font color="Maroon" title="c8">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">x</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c9">x</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">x</font>)</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c10">a1</font> = <font color="Maroon" title="c8">a</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" onmouseover="rs('xcmplx_0/D2')" onmouseout="rh()">XCMPLX_0:def 2</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c8">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x</font> = 
 the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">a1</font>,<font color="Maroon" title="c9">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
 the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">x</font>)

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E7">A5</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>)</span><br/><span class="kw">thus </span><font color="Maroon" title="c8">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">y</font> = 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>)

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:7"/>
(  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c8">Mv</font> =  the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">Av</font> =  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>;<br/>
<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D2" onmouseover="rs('rlvect_1/D2')" onmouseout="rh()">RLVECT_1:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">x</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">y1</font> = <font color="Maroon" title="c11">y</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">y</font> = 
<font color="Maroon" title="c12">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">y1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">x1</font>

<br/>.= 
<font color="Maroon" title="c11">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_3_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font>, <font color="Maroon" title="c12">z</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">z</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c14">y1</font> = <font color="Maroon" title="c11">y</font>, <font color="Maroon" title="c15">z1</font> = <font color="Maroon" title="c12">z</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">z</font> = 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">y</font></span>)</span>,<font color="Maroon" title="c15">z1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">y1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">z1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">z1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">z</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:7_3"/>
 <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_3_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">x</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #)</span>)</span> = 
<font color="Maroon" title="c11">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:7_3"/>
 <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )</span><br/>

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="clvect_1.html#D2" onmouseover="rs('clvect_1/D2')" onmouseout="rh()">CLVECT_1:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c10">a1</font> be   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) holds  <font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">y</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c10">a1</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" onmouseover="rs('xcmplx_0/D2')" onmouseout="rh()">XCMPLX_0:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c15">y1</font> = <font color="Maroon" title="c12">y</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">y</font></span>)</span> = 
 the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">y</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x1</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="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D2" onmouseover="rs('clvect_1/D2')" onmouseout="rh()">CLVECT_1:def 2</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c11">x</font>)</span>)</span>,<span class="p1">(<span class="default"> the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c12">y</font>)</span>)</span>)

<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c11">x</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">y</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">y</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E4:7_3_4"/>
<font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">y</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="clvect_1.html#D3" onmouseover="rs('clvect_1/D3')" onmouseout="rh()">CLVECT_1:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c10">a1</font>, <font color="Maroon" title="c11">b1</font> be   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c12">x</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c10">a1</font>, <font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c11">b1</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" onmouseover="rs('xcmplx_0/D2')" onmouseout="rh()">XCMPLX_0:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c12">x</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font> = 
 the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">b</font></span>)</span>,<font color="Maroon" title="c12">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y</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="c14">b</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D3" onmouseover="rs('clvect_1/D3')" onmouseout="rh()">CLVECT_1:def 3</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"> the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c15">y</font>)</span>)</span>,<span class="p1">(<span class="default"> the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c15">y</font>)</span>)</span>)

<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c12">x</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E4:7_3_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="clvect_1.html#D4" onmouseover="rs('clvect_1/D4')" onmouseout="rh()">CLVECT_1:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a> </span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c10">a1</font>, <font color="Maroon" title="c11">b1</font> be   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c12">x</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c10">a1</font>, <font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c11">b1</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" onmouseover="rs('xcmplx_0/D2')" onmouseout="rh()">XCMPLX_0:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c12">x</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font> = 
 the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">b</font></span>)</span>,<font color="Maroon" title="c12">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y</font>

<br/>.= 
<font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D4" onmouseover="rs('clvect_1/D4')" onmouseout="rh()">CLVECT_1:def 4</a></span>
<br/>.= 
 the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c13">a</font>,<span class="p1">(<span class="default"> the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c15">y</font>)</span>)</span>)

<br/>.= 
 the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E4:7_3_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c10">x</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>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="clvect_1.html#D5" onmouseover="rs('clvect_1/D5')" onmouseout="rh()">CLVECT_1:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">x</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">thus </span><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c10">x</font> = 
 the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>,<font color="Maroon" title="c10">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>
<br/>.= 
<a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c11">y</font>

<br/>.= 
<font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D5" onmouseover="rs('clvect_1/D5')" onmouseout="rh()">CLVECT_1:def 5</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:7"/>
(  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">M</font> #) is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
