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<b>let </b><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="rlvect_1.html#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">RLSStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K1" title="NUMBERS:func.1">REAL</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="rlvect_1.html#U1" title="RLVECT_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#K1" title="NUMBERS:func.1">REAL</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> )</span><br/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K1" title="NUMBERS:func.1">REAL</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="rlvect_1.html#U1" title="RLVECT_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#K1" title="NUMBERS:func.1">REAL</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> )</span><br/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K1" title="NUMBERS:func.1">REAL</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="rlvect_1.html#U1" title="RLVECT_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#K1" title="NUMBERS:func.1">REAL</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> )</span><br/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K1" title="NUMBERS:func.1">REAL</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="rlvect_1.html#U1" title="RLVECT_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#K1" title="NUMBERS:func.1">REAL</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> )</span><br/><b>let </b><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#K1" title="NUMBERS:func.1">REAL</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="rlvect_1.html#U1" title="RLVECT_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#K1" title="NUMBERS:func.1">REAL</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> ) )</span><br/>









<b>set </b><font color="Maroon" title="c6">D</font> = <font color="Maroon" title="c2">V1</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:5"/><i><font color="Green" title="E3">A2</font></i>: 
<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>
 <b>and </b><br/><a NAME="E2:5"/><i><font color="Green" title="E4">A3</font></i>: 
<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>
 <b>and </b><br/><a NAME="E3:5"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_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#K1" title="NUMBERS:func.1">REAL</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> )</span><br/>

<b>set </b><font color="Maroon" title="c7">W</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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><i><font color="Green" title="E6">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>thus </b><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>
<b>by </b><i><a class="txt" href="#E2:5"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E7">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_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/><b>let </b><font color="Maroon" title="c9">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_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/><b>thus </b><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x</font> = 
the <a href="rlvect_1.html#U1" title="RLVECT_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="c8">a</font>,<font color="Maroon" title="c9">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="#E3:5"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E6:5"/>
(  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c8">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/>
<b>set </b><font color="Maroon" title="c9">Mv</font> = the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> )</span>
<div class="add"><b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>reconsider </b><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="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>thus </b><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>
<b>by </b><i><a class="txt" href="#E4:5"><i><font color="Green" title="E6">A6</font></i></a></i>
<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>
<b>by </b><i><a class="txt" href="#E4:5"><i><font color="Green" title="E6">A6</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font>, <font color="Maroon" title="c12">z</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>reconsider </b><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="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>thus </b><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>
<b>by </b><i><a class="txt" href="#E4:5"><i><font color="Green" title="E6">A6</font></i></a></i>
<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>
<b>by </b><i><a class="txt" href="#E4:5"><i><font color="Green" title="E6">A6</font></i></a></i>
<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>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<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>
<b>by </b><i><a class="txt" href="#E4:5"><i><font color="Green" title="E6">A6</font></i></a></i>
<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>
<b>by </b><i><a class="txt" href="#E4:5"><i><font color="Green" title="E6">A6</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:5_3"/>
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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/><b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>thus </b><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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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>
<b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="#E4:5"><i><font color="Green" title="E6">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:5_3"/>
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a></span><br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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">b<sub>1</sub></font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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">b<sub>1</sub></font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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  1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) )</span>
<div class="add"><b>let </b><font color="Maroon" title="c10">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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">a</font> <a href="rlvect_1.html#K1" title="RLVECT_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">a</font> <a href="rlvect_1.html#K1" title="RLVECT_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">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_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">a</font> <a href="rlvect_1.html#K1" title="RLVECT_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">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">y</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c14">y1</font> = <font color="Maroon" title="c12">y</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>thus </b><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_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="rlvect_1.html#U1" title="RLVECT_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="c10">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>
<b>by </b><i><a class="txt" href="#E5:5"><i><font color="Green" title="E7">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <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>
<b>by </b><i><a class="txt" href="#E4:5"><i><font color="Green" title="E6">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">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="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<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="rlvect_1.html#U1" title="RLVECT_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="c10">a</font>,<font color="Maroon" title="c13">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">y</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:5"><i><font color="Green" title="E7">A7</font></i></a></i>
<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="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">y</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:5"><i><font color="Green" title="E7">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_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">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">y</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:5"><i><font color="Green" title="E6">A6</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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">b<sub>1</sub></font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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  1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) )</span>
<div class="add"><b>let </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_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">a</font> <a href="rlvect_1.html#K1" title="RLVECT_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">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c12">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_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">a</font> <a href="rlvect_1.html#K1" title="RLVECT_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">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="txt" href="#E5:5"><i><font color="Green" title="E7">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">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="c11">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<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="rlvect_1.html#U1" title="RLVECT_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="c10">a</font>,<font color="Maroon" title="c13">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:5"><i><font color="Green" title="E7">A7</font></i></a></i>
<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="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:5"><i><font color="Green" title="E7">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_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">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:5"><i><font color="Green" title="E6">A6</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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  1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font></span>
<div class="add"><b>let </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_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">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c12">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_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">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="txt" href="#E5:5"><i><font color="Green" title="E7">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_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="c10">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:5"><i><font color="Green" title="E7">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:5"><i><font color="Green" title="E7">A7</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c10">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 1 <a href="rlvect_1.html#K1" title="RLVECT_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/>

<b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b>1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">x</font> = 
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">y</font>
<b>by </b><i><a class="txt" href="#E5:5"><i><font color="Green" title="E7">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:5"/>
(  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</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#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
