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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">d1</font>, <font color="Olive" title="b2">d2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</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="Maroon" title="c1">X</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"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b5">V</font> being  <a href="funcsdom.html#NM2" title="FUNCSDOM:NM.2">Algebra</a><br/> for <font color="Olive" title="b6">V1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b5">V</font><br/> for <font color="Olive" title="b7">MR</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="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b6">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <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="Olive" title="b5">V</font> &amp; <font color="Olive" title="b2">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b5">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="Olive" title="b5">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b6">V1</font> &amp; <font color="Olive" title="b4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b5">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b6">V1</font> &amp; <font color="Olive" title="b7">MR</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="Olive" title="b5">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="b6">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Olive" title="b6">V1</font> is <a href="c0sp1.html#V1" title="C0SP1:attr.1">having-inverse</a> holds <br/> <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">M</font>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b7">MR</font>,<font color="Olive" title="b2">d2</font>,<font color="Olive" title="b1">d1</font> #) is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Olive" title="b5">V</font></span><br/><b>let </b><font color="Maroon" title="c2">d1</font>, <font color="Maroon" title="c3">d2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</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="c1">X</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"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">V</font> being  <a href="funcsdom.html#NM2" title="FUNCSDOM:NM.2">Algebra</a><br/> for <font color="Olive" title="b4">V1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b3">V</font><br/> for <font color="Olive" title="b5">MR</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="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b4">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">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="Olive" title="b3">V</font> &amp; <font color="Maroon" title="c3">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b3">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="Olive" title="b3">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b4">V1</font> &amp; <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b3">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b4">V1</font> &amp; <font color="Olive" title="b5">MR</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="Olive" title="b3">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="b4">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Olive" title="b4">V1</font> is <a href="c0sp1.html#V1" title="C0SP1:attr.1">having-inverse</a> holds <br/> <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b5">MR</font>,<font color="Maroon" title="c3">d2</font>,<font color="Maroon" title="c2">d1</font> #) is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Olive" title="b3">V</font></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="c1">X</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"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">V</font> being  <a href="funcsdom.html#NM2" title="FUNCSDOM:NM.2">Algebra</a><br/> for <font color="Olive" title="b3">V1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">V</font><br/> for <font color="Olive" title="b4">MR</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="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">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="Olive" title="b2">V</font> &amp; <font color="Maroon" title="c3">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b2">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="Olive" title="b2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b3">V1</font> &amp; <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b3">V1</font> &amp; <font color="Olive" title="b4">MR</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="Olive" title="b2">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="b3">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Olive" title="b3">V1</font> is <a href="c0sp1.html#V1" title="C0SP1:attr.1">having-inverse</a> holds <br/> <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">M</font>,<font color="Maroon" title="c4">A</font>,<font color="Olive" title="b4">MR</font>,<font color="Maroon" title="c3">d2</font>,<font color="Maroon" title="c2">d1</font> #) is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Olive" title="b2">V</font></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"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">X</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">V</font> being  <a href="funcsdom.html#NM2" title="FUNCSDOM:NM.2">Algebra</a><br/> for <font color="Olive" title="b2">V1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b3">MR</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="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">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="Olive" title="b1">V</font> &amp; <font color="Maroon" title="c3">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b1">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="Olive" title="b1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b2">V1</font> &amp; <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b2">V1</font> &amp; <font color="Olive" title="b3">MR</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="Olive" title="b1">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="b2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Olive" title="b2">V1</font> is <a href="c0sp1.html#V1" title="C0SP1:attr.1">having-inverse</a> holds <br/> <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c4">A</font>,<font color="Olive" title="b3">MR</font>,<font color="Maroon" title="c3">d2</font>,<font color="Maroon" title="c2">d1</font> #) is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Olive" title="b1">V</font></span><br/><b>let </b><font color="Maroon" title="c6">V</font> be   <a href="funcsdom.html#NM2" title="FUNCSDOM:NM.2">Algebra</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  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">V</font><br/> for <font color="Olive" title="b2">MR</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="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">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="c6">V</font> &amp; <font color="Maroon" title="c3">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c6">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="c6">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">V1</font> &amp; <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c6">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">V1</font> &amp; <font color="Olive" title="b2">MR</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="c6">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> &amp; <font color="Olive" title="b1">V1</font> is <a href="c0sp1.html#V1" title="C0SP1:attr.1">having-inverse</a> holds <br/> <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c4">A</font>,<font color="Olive" title="b2">MR</font>,<font color="Maroon" title="c3">d2</font>,<font color="Maroon" title="c2">d1</font> #) is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Maroon" title="c6">V</font></span><br/><b>let </b><font color="Maroon" title="c7">V1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">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">MR</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="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c7">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">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="c6">V</font> &amp; <font color="Maroon" title="c3">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c6">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="c6">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">V1</font> &amp; <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c6">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">V1</font> &amp; <font color="Olive" title="b1">MR</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="c6">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="c7">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c7">V1</font> is <a href="c0sp1.html#V1" title="C0SP1:attr.1">having-inverse</a> holds <br/> <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c4">A</font>,<font color="Olive" title="b1">MR</font>,<font color="Maroon" title="c3">d2</font>,<font color="Maroon" title="c2">d1</font> #) is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Maroon" title="c6">V</font></span><br/><b>let </b><font color="Maroon" title="c8">MR</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="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">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="c6">V</font> &amp; <font color="Maroon" title="c3">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c6">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="c6">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">V1</font> &amp; <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c6">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">V1</font> &amp; <font color="Maroon" title="c8">MR</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="c6">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="c7">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c7">V1</font> is <a href="c0sp1.html#V1" title="C0SP1:attr.1">having-inverse</a> implies  <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c8">MR</font>,<font color="Maroon" title="c3">d2</font>,<font color="Maroon" title="c2">d1</font> #) is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Maroon" title="c6">V</font> )</span><br/>















<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c7">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c2">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="c6">V</font>
 <b>and </b><br/><a NAME="E3:18"/><i><font color="Green" title="E16">B2</font></i>: 
<font color="Maroon" title="c3">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c6">V</font>
 <b>and </b><br/><a NAME="E4:18"/><i><font color="Green" title="E17">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="c6">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">V1</font>
 <b>and </b><br/><a NAME="E5:18"/><i><font color="Green" title="E18">B3</font></i>: 
<font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c6">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">V1</font>
 <b>and </b><br/><a NAME="E6:18"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c8">MR</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="c6">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="c7">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E7:18"/><i><font color="Green" title="E20">B4</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">V</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font> holds <br/> <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="c0sp1.html#D1" target="_self" title="C0SP1:def.1">C0SP1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c8">MR</font>,<font color="Maroon" title="c3">d2</font>,<font color="Maroon" title="c2">d1</font> #) is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Maroon" title="c6">V</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c9">W</font> =  <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c8">MR</font>,<font color="Maroon" title="c3">d2</font>,<font color="Maroon" title="c2">d1</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="funcsdom.html#L1" title="FUNCSDOM:struct.1">AlgebraStr</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E9:18"/><i><font color="Green" title="E21">A5</font></i>: 
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c9">W</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="c6">V</font> &amp;  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c9">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c6">V</font> )
 
<b>by </b><i><a class="txt" href="c0sp1.html#E2:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="c0sp1.html#E3:18"><i><font color="Green" title="E16">B2</font></i></a></i>;<br/>
<div><i><font color="Green" title="E22">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">W</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> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c6">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><br/><a NAME="E1:18_1"/>
<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> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c6">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">x</font>,<font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="c0sp1.html#E4:18"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E2:18_1"/>
<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> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c6">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">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="E23">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><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> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c9">W</font> holds  <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> <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="c6">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Olive" title="b1">x</font></span><br/><b>let </b><font color="Maroon" title="c11">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c9">W</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> <font color="Maroon" title="c11">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="c6">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">x</font></span><br/><a NAME="E1:18_2"/>
<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> <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="c6">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">a</font>,<font color="Maroon" title="c11">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="c0sp1.html#E6:18"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E2:18_2"/>
<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> <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="c6">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">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>
<div><i><font color="Green" title="E24">B7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c9">W</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c6">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">x</font></span><br/><a NAME="E1:18_3"/>
<font color="Maroon" title="c10">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c6">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">a</font>,<font color="Maroon" title="c11">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="c0sp1.html#E5:18"><i><font color="Green" title="E18">B3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E2:18_3"/>
<font color="Maroon" title="c10">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c6">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">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="E13:18"/>
( <font color="Maroon" title="c9">W</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">Algebra-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c10">Av</font> = the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c6">V</font>;<br/>
<b>set </b><font color="Maroon" title="c11">MV</font> = the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c6">V</font>;<br/>
<b>set </b><font color="Maroon" title="c12">Mv</font> = the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c6">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"> ( <font color="Maroon" title="c9">W</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">Algebra-like</a> )</span>
<div class="add"><b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c9">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">x</font></span><br/><b>reconsider </b><font color="Maroon" title="c15">x1</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c16">y1</font> = <font color="Maroon" title="c14">y</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c6">V</font> <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E2:18_4_1"/>
( <font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">y1</font> &amp; <font color="Maroon" title="c14">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">x1</font> )
 <b>by </b><i><a class="txt" href="c0sp1.html#E10:18"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:18_4_1"/>
<font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">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>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">W</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">Algebra-like</a> )</span>
<div class="add"><b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font>, <font color="Maroon" title="c15">z</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c9">W</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="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c16">x1</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c17">y1</font> = <font color="Maroon" title="c14">y</font>, <font color="Maroon" title="c18">z1</font> = <font color="Maroon" title="c15">z</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c6">V</font> <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E2:18_4_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</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="c6">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">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font></span>)</span>,<font color="Maroon" title="c18">z1</font> &amp; <font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</font></span>)</span> <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="c6">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">x1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</font></span>)</span> )
 <b>by </b><i><a class="txt" href="c0sp1.html#E10:18"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/><a NAME="E3:18_4_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">y1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">z1</font> &amp; <font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">z1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="c0sp1.html#E10:18"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E4:18_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</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>; <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> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">W</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">Algebra-like</a> )</span>
<div class="add"><b>let </b><font color="Maroon" title="c13">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c9">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">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> <font color="Maroon" title="c9">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font></span><br/><b>reconsider </b><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c15">z</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c9">W</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c6">V</font> <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>thus </b><font color="Maroon" title="c13">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> <font color="Maroon" title="c9">W</font></span>)</span> = 
<font color="Maroon" title="c14">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="c6">V</font></span>)</span>
<b>by </b><i><a class="txt" href="c0sp1.html#E2:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="c0sp1.html#E10:18"><i><font color="Green" title="E22">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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>thus </b><a NAME="E4:18_4"/>
<font color="Maroon" title="c9">W</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">W</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">Algebra-like</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c13">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">V</font> <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">V</font><b> such that </b><br/><a NAME="E3:18_4_4"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c14">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">v</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="c6">V</font>
 <b>by </b><i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a></i>;<br/>
<a NAME="E4:18_4_4"/>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c14">x1</font>
 
<b>by </b><i><a class="txt" href="c0sp1.html#E3:18_4_4"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="vectsp_1.html#T63" title="VECTSP_1:th.63">VECTSP_1:63</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c15">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">W</font> <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="c0sp1.html#E7:18"><i><font color="Green" title="E20">B4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">y</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c16">y</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="c9">W</font></span><br/>

<b>thus </b><a NAME="E6:18_4_4"/>
<font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c16">y</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="c9">W</font>
 <b>by </b><i><a class="txt" href="c0sp1.html#E2:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="c0sp1.html#E10:18"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="c0sp1.html#E3:18_4_4"><i><font color="Green" title="E25">A8</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> <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">W</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">Algebra-like</a> )</span>
<div class="add"><b>let </b><font color="Maroon" title="c13">v</font>, <font color="Maroon" title="c14">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">v</font></span><br/><b>reconsider </b><font color="Maroon" title="c15">v1</font> = <font color="Maroon" title="c13">v</font>, <font color="Maroon" title="c16">w1</font> = <font color="Maroon" title="c14">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">V</font> <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E2:18_4_5"/>
( <font color="Maroon" title="c13">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c16">w1</font> &amp; <font color="Maroon" title="c14">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">w1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c15">v1</font> )
 <b>by </b><i><a class="txt" href="c0sp1.html#E12:18"><i><font color="Green" title="E24">B7</font></i></a></i>;<br/><b>hence </b><a NAME="E3:18_4_5"/>
<font color="Maroon" title="c13">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">v</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><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">Algebra-like</a> )</span>
<div class="add"><b>let </b><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font>, <font color="Maroon" title="c15">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">W</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="c13">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">x</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c17">a1</font> = <font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c18">b1</font> = <font color="Maroon" title="c14">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">V</font> <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E2:18_4_6"/>
( <font color="Maroon" title="c13">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c18">b1</font> &amp; <font color="Maroon" title="c13">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c6">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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">x</font></span>)</span> )
 <b>by </b><i><a class="txt" href="c0sp1.html#E12:18"><i><font color="Green" title="E24">B7</font></i></a></i>;<br/><a NAME="E3:18_4_6"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c18">b1</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c16">y</font> &amp; <font color="Maroon" title="c13">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c16">y</font></span>)</span> )
 <b>by </b><i><a class="txt" href="c0sp1.html#E12:18"><i><font color="Green" title="E24">B7</font></i></a></i>;<br/><b>hence </b><a NAME="E4:18_4_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</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> <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">W</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">Algebra-like</a> )</span>
<div class="add"><b>let </b><font color="Maroon" title="c13">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c9">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font></span><br/><b>reconsider </b><font color="Maroon" title="c14">v1</font> = <font color="Maroon" title="c13">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">V</font> <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E2:18_4_7"/>
<font color="Maroon" title="c13">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c9">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c6">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="c0sp1.html#E12:18"><i><font color="Green" title="E24">B7</font></i></a>, <a class="txt" href="c0sp1.html#E3:18"><i><font color="Green" title="E16">B2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:18_4_7"/>
<font color="Maroon" title="c13">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c9">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</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> <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">W</font> is <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">Algebra-like</a></span>
<div class="add"><b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font>, <font color="Maroon" title="c15">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</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">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">z</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c16">x1</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c17">y1</font> = <font color="Maroon" title="c14">y</font>, <font color="Maroon" title="c18">z1</font> = <font color="Maroon" title="c15">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">V</font> <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E2:18_4_8"/><i><font color="Green" title="E25">T2</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c17">y1</font> &amp; <font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c18">z1</font> )
 <b>by </b><i><a class="txt" href="c0sp1.html#E12:18"><i><font color="Green" title="E24">B7</font></i></a></i>;<br/><a NAME="E3:18_4_8"/>
<font color="Maroon" title="c14">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">z1</font>
 <b>by </b><i><a class="txt" href="c0sp1.html#E10:18"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/><a NAME="E4:18_4_8"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">z1</font></span>)</span>
 <b>by </b><i><a class="txt" href="c0sp1.html#E12:18"><i><font color="Green" title="E24">B7</font></i></a></i>;<br/><a NAME="E5:18_4_8"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c17">y1</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="c16">x1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c18">z1</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a></i>;<br/><b>hence </b><a NAME="E6:18_4_8"/>
<font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</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">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="c0sp1.html#E2:18_4_8"><i><font color="Green" title="E25">T2</font></i></a>, <a class="txt" href="c0sp1.html#E10:18"><i><font color="Green" title="E22">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>thus </b><a NAME="E9:18_4"/>
<font color="Maroon" title="c9">W</font> is <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">Algebra-like</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font>, <font color="Maroon" title="c15">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="funcsdom.html#D20" title="FUNCSDOM:def.20">FUNCSDOM:def 20</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>  holds <br/> ( <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="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <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="Maroon" title="c13">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font> &amp; <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="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font></span>)</span> <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="Maroon" title="c13">x</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>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</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="Maroon" title="c13">x</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="Maroon" title="c13">x</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="Maroon" title="c13">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</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="Maroon" title="c13">x</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="Maroon" title="c13">x</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c16">a</font>, <font color="Maroon" title="c17">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"> ( <font color="Maroon" title="c16">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">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font> &amp; <font color="Maroon" title="c16">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">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">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="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">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="c17">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font></span>)</span> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c18">x1</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c19">y1</font> = <font color="Maroon" title="c14">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">V</font> <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E2:18_4_9"/><i><font color="Green" title="E25">T0</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">y1</font>
 
<b>by </b><i><a class="txt" href="c0sp1.html#E10:18"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/>
<a NAME="E3:18_4_9"/><i><font color="Green" title="E26">T4</font></i>: 
( <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">x1</font> &amp; <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">y1</font> &amp; <font color="Maroon" title="c17">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">x1</font> )
 
<b>by </b><i><a class="txt" href="c0sp1.html#E11:18"><i><font color="Green" title="E23">A7</font></i></a></i>;<br/>
<a NAME="E4:18_4_9"/>
<font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c19">y1</font>
 
<b>by </b><i><a class="txt" href="c0sp1.html#E12:18"><i><font color="Green" title="E24">B7</font></i></a></i>;<br/>
<a NAME="E5:18_4_9"/><b>then </b>
<font color="Maroon" title="c16">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">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c19">y1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="c0sp1.html#E11:18"><i><font color="Green" title="E23">A7</font></i></a></i>;<br/>
<a NAME="E6:18_4_9"/><b>then </b>
<font color="Maroon" title="c16">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">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">x1</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c19">y1</font>
 
<b>by </b><i><a class="ref" href="funcsdom.html#D20" title="FUNCSDOM:def.20">FUNCSDOM:def 20</a></i>;<br/>
<b>hence </b><a NAME="E7:18_4_9"/>
<font color="Maroon" title="c16">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">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font>
 <b>by </b><i><a class="txt" href="c0sp1.html#E3:18_4_9"><i><font color="Green" title="E26">T4</font></i></a>, <a class="txt" href="c0sp1.html#E12:18"><i><font color="Green" title="E24">B7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">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">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">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="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">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="c17">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font></span>)</span> )</span><br/>

<a NAME="E8:18_4_9"/>
<font color="Maroon" title="c16">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">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">y1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="c0sp1.html#E2:18_4_9"><i><font color="Green" title="E25">T0</font></i></a>, <a class="txt" href="c0sp1.html#E11:18"><i><font color="Green" title="E23">A7</font></i></a></i>;<br/>
<a NAME="E9:18_4_9"/><b>then </b>
<font color="Maroon" title="c16">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">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">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="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">y1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funcsdom.html#D20" title="FUNCSDOM:def.20">FUNCSDOM:def 20</a></i>;<br/>
<b>hence </b><a NAME="E10:18_4_9"/>
<font color="Maroon" title="c16">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">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">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="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="c0sp1.html#E3:18_4_9"><i><font color="Green" title="E26">T4</font></i></a>, <a class="txt" href="c0sp1.html#E10:18"><i><font color="Green" title="E22">A6</font></i></a></i>; <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="c16">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">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="c17">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font></span>)</span> )</span><br/>

<a NAME="E11:18_4_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">x1</font>
 
<b>by </b><i><a class="txt" href="c0sp1.html#E11:18"><i><font color="Green" title="E23">A7</font></i></a></i>;<br/>
<a NAME="E12:18_4_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">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="c17">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">x1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funcsdom.html#D20" title="FUNCSDOM:def.20">FUNCSDOM:def 20</a></i>;<br/>
<b>hence </b><a NAME="E13:18_4_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">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="c17">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="c0sp1.html#E3:18_4_9"><i><font color="Green" title="E26">T4</font></i></a>, <a class="txt" href="c0sp1.html#E10:18"><i><font color="Green" title="E22">A6</font></i></a></i>; <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="c16">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span><br/>

<a NAME="E14:18_4_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">x1</font>
 
<b>by </b><i><a class="txt" href="c0sp1.html#E11:18"><i><font color="Green" title="E23">A7</font></i></a></i>;<br/>
<a NAME="E15:18_4_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">x1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funcsdom.html#D20" title="FUNCSDOM:def.20">FUNCSDOM:def 20</a></i>;<br/>
<b>hence </b><a NAME="E16:18_4_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="c0sp1.html#E11:18"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="c0sp1.html#E3:18_4_9"><i><font color="Green" title="E26">T4</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>end;</b></div>
<b>hence </b><a NAME="E14:18"/>
 <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c8">MR</font>,<font color="Maroon" title="c3">d2</font>,<font color="Maroon" title="c2">d1</font> #) is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Maroon" title="c6">V</font>
 <b>by </b><i><a class="txt" href="c0sp1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="c0sp1.html#E4:18"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="c0sp1.html#E5:18"><i><font color="Green" title="E18">B3</font></i></a>, <a class="txt" href="c0sp1.html#E6:18"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="c0sp1.html#E9:18"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="c0sp1.html#D9" target="_self" title="C0SP1:def.9">DefSubAlg</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>
