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<span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b2">D</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">D</font><br/>  for <font color="Olive" title="b4">A</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b2">D</font><br/>  for <font color="Olive" title="b5">M</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Olive" title="b2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Olive" title="b2">D</font><br/>  for <font color="Olive" title="b6">S</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="Olive" title="b2">D</font>,<font color="Olive" title="b2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">D</font> &amp; <font color="Olive" title="b3">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="c2">V</font> &amp; <font color="Olive" title="b4">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="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">V1</font> &amp; <font color="Olive" title="b5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</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="b6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">V1</font> holds <br/> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Olive" title="b2">D</font>,<font color="Olive" title="b3">d1</font>,<font color="Olive" title="b4">A</font>,<font color="Olive" title="b5">M</font>,<font color="Olive" title="b6">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">V1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">D</font><br/>  for <font color="Olive" title="b3">A</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b1">D</font><br/>  for <font color="Olive" title="b4">M</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Olive" title="b1">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Olive" title="b1">D</font><br/>  for <font color="Olive" title="b5">S</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="Olive" title="b1">D</font>,<font color="Olive" title="b1">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Olive" title="b4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</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="c3">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Olive" title="b5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> holds <br/> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Olive" title="b1">D</font>,<font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">M</font>,<font color="Olive" title="b5">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">D</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">D</font><br/>  for <font color="Olive" title="b2">A</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c4">D</font><br/>  for <font color="Olive" title="b3">M</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Maroon" title="c4">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">D</font><br/>  for <font color="Olive" title="b4">S</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="c4">D</font>,<font color="Maroon" title="c4">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</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="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Olive" title="b3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</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="c3">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Olive" title="b4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> holds <br/> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">M</font>,<font color="Olive" title="b4">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">d1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c4">D</font><br/>  for <font color="Olive" title="b2">M</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Maroon" title="c4">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">D</font><br/>  for <font color="Olive" title="b3">S</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="c4">D</font>,<font color="Maroon" title="c4">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c5">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="c2">V</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</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="c3">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Olive" title="b3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> holds <br/> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">A</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c4">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Maroon" title="c4">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">D</font><br/>  for <font color="Olive" title="b2">S</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="c4">D</font>,<font color="Maroon" title="c4">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c5">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="c2">V</font> &amp; <font color="Maroon" title="c6">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="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</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="c3">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> holds <br/> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">M</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Maroon" title="c4">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</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="c4">D</font>,<font color="Maroon" title="c4">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c5">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="c2">V</font> &amp; <font color="Maroon" title="c6">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="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Maroon" title="c7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</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="c3">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> holds <br/> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Olive" title="b1">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">S</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="c4">D</font>,<font color="Maroon" title="c4">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c5">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="c2">V</font> &amp; <font color="Maroon" title="c6">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="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Maroon" title="c7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</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="c3">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c8">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> implies  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:18"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font>
 <span class="kw">and </span><br/><a NAME="E2:18"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c5">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="c2">V</font>
 <span class="kw">and </span><br/><a NAME="E3:18"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c6">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="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font>
 <span class="kw">and </span><br/><a NAME="E4:18"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</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="c3">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:18"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<font color="Maroon" title="c8">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font></span><br/>

<a NAME="E6:18"/>
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c9">W</font> =  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #);<br/>
<a NAME="E1:18_1"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">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="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</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="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c11">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">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="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c12">a</font> = <font color="Maroon" title="c10">a</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E2:18_1_1"/>
<font color="Maroon" title="c12">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="c2">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="c12">a</font>,<font color="Maroon" title="c11">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:18_1_1"/>
<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="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:18_1"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) holds  <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E3:18_1"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E4:18_1"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
(  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="bhsp_1.html#V2" title="BHSP_1:attr.2">RealUnitarySpace-like</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c10">SV</font> =  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font>;<br/>
<span class="kw">set </span><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="c2">V</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">AV</font> =  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V</font>;<br/>
<a NAME="E1:18_1_2"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) holds  <font color="Maroon" title="c1">jj</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> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">jj</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="c13">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c13">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c1">jj</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font> = 
<font color="Maroon" title="c1">jj</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" onmouseover="rs('rlvect_1/D8')" onmouseout="rh()">RLVECT_1:def 8</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:18_1_2"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b3">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">x</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c15">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c15">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">a</font> = <font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c18">b</font> = <font color="Maroon" title="c14">b</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">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">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" onmouseover="rs('rlvect_1/D7')" onmouseout="rh()">RLVECT_1:def 7</a></span>
<br/>.= 
 the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">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="c17">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">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">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:18_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:18_1_2"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">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" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">y</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) holds  <font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c15">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c16">x1</font> = <font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c17">y1</font> = <font color="Maroon" title="c15">y</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">a</font> = <font color="Maroon" title="c13">a</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<font color="Maroon" title="c18">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">y</font></span>)</span> = 
 the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">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="c18">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">y</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c18">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">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="c18">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c17">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D5" onmouseover="rs('rlvect_1/D5')" onmouseout="rh()">RLVECT_1:def 5</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c16">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><font color="Maroon" title="c18">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">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="c18">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:18_1_2_3"/>
<font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:18_1_2"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c13">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="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> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</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="c2">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" onmouseover="rs('rlvect_1/T4')" onmouseout="rh()">RLVECT_1:4</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E5:18_1_2"/>
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="bhsp_1.html#V2" title="BHSP_1:attr.2">RealUnitarySpace-like</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="bhsp_1.html#D2" onmouseover="rs('bhsp_1/D2')" onmouseout="rh()">BHSP_1:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( (  not <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) ) &amp; (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) or <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_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="c14">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> &amp; <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="bhsp_1.html#K1" title="BHSP_1: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="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span> &amp; <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="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">y</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c16">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="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">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="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">x1</font> = <font color="Maroon" title="c13">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c19">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  not <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) ) &amp; (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) or <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_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="c14">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> &amp; <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="bhsp_1.html#K1" title="BHSP_1: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="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c19">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="bhsp_1.html#K1" title="BHSP_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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">y</font></span>)</span> )</span><br/>

<a NAME="E4:18_1_2_5"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) implies <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_2_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_1_2_5_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E2:18_1_2_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c18">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="bhsp_1.html#D2" onmouseover="rs('bhsp_1/D2')" onmouseout="rh()">BHSP_1:def 2</a></span>;<br/>
<a NAME="E3:18_1_2_5_1"/><span class="kw">then </span>
 the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c18">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E4:18_1_2_5_1"/>
<font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:18_1_2_5"/>
( <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_2_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_1_2_5_2"/>
<font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #)</span><br/>

<a NAME="E2:18_1_2_5_2"/><span class="kw">then </span>
 the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c18">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a></span>;<br/>
<a NAME="E3:18_1_2_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c18">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E4:18_1_2_5_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="bhsp_1.html#D2" onmouseover="rs('bhsp_1/D2')" onmouseout="rh()">BHSP_1:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:18_1_2_5"/>
( <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  iff <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_1_2_5"><span class="lab"><font color="Green" title="E28">A14</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_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="c14">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> &amp; <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="bhsp_1.html#K1" title="BHSP_1: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="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c19">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="bhsp_1.html#K1" title="BHSP_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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">y</font></span>)</span> )</span><br/>

<a NAME="E7:18_1_2_5"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c18">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_1.html#D2" onmouseover="rs('bhsp_1/D2')" onmouseout="rh()">BHSP_1:def 2</a></span>;<br/>
<a NAME="E8:18_1_2_5"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c18">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
;<br/>
<span class="kw">hence </span><a NAME="E9:18_1_2_5"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_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="c14">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font> &amp; <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="bhsp_1.html#K1" title="BHSP_1: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="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c19">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="bhsp_1.html#K1" title="BHSP_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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">y</font></span>)</span> )</span><br/>

<a NAME="E10:18_1_2_5"/>
 the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c17">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c18">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_1.html#D1" onmouseover="rs('bhsp_1/D1')" onmouseout="rh()">BHSP_1:def 1</a></span>;<br/>
<a NAME="E11:18_1_2_5"/><span class="kw">then </span>
 the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c17">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">y1</font>,<font color="Maroon" title="c18">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
;<br/>
<a NAME="E12:18_1_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">y1</font>,<font color="Maroon" title="c18">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:18_1_2_5"/>
<font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_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="c14">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="bhsp_1.html#K1" title="BHSP_1: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="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c19">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="bhsp_1.html#K1" title="BHSP_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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">y</font></span>)</span> )</span><br/>

<span class="lab"><font color="Green" title="E29">A15</font></span>: <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="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font> = 
 the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">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"><span class="p2">(<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="c15">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a></span>
<br/>.= 
 the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">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"><span class="p2">(<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="c17">y1</font></span>)</span>,<font color="Maroon" title="c15">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>
<br/>.= 
<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="c17">y1</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c16">z1</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c16">z1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">y1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c16">z1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_1.html#D2" onmouseover="rs('bhsp_1/D2')" onmouseout="rh()">BHSP_1:def 2</a></span>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span> = 
<span class="p1">(<span class="default"> the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c15">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c15">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"> the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c15">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c16">z1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">y1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c16">z1</font></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E16:18_1_2_5"/>
<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="bhsp_1.html#K1" title="BHSP_1: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="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c15">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18_1_2_5"><span class="lab"><font color="Green" title="E29">A15</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c19">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="bhsp_1.html#K1" title="BHSP_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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">y</font></span>)</span></span><br/>

<span class="lab"><font color="Green" title="E30">A16</font></span>: <font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">y</font></span>)</span> = 
<font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"> the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c17">y1</font></span>)</span>

;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">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="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">y</font> = 
 the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font></span>)</span>,<font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a></span>
<br/>.= 
 the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">x1</font></span>)</span>,<font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">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="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c17">y1</font>

<br/>.= 
<font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c17">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_1.html#D2" onmouseover="rs('bhsp_1/D2')" onmouseout="rh()">BHSP_1:def 2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E19:18_1_2_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">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="bhsp_1.html#K1" title="BHSP_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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:18_1_2_5"><span class="lab"><font color="Green" title="E30">A16</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E6:18_1_2"/><span class="lab"><font color="Green" title="E28">A17</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b3">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">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" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">x</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c15">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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="c13">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c15">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">a</font> = <font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c18">b</font> = <font color="Maroon" title="c14">b</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" onmouseover="rs('rlvect_1/D6')" onmouseout="rh()">RLVECT_1:def 6</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c16">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">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="c18">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:18_1_2_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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="c13">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">x</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:18_1_2"/><span class="lab"><font color="Green" title="E29">A18</font></span>: 
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D16" onmouseover="rs('algstr_0/D16')" onmouseout="rh()">ALGSTR_0:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c13">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E3:18_1_2_7"/><span class="lab"><font color="Green" title="E30">A19</font></span>: 
<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="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_0.html#D11" onmouseover="rs('algstr_0/D11')" onmouseout="rh()">ALGSTR_0:def 11</a></span>;<br/>
<font color="Maroon" title="c15">v</font> = 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c14">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1_2_7"><span class="lab"><font color="Green" title="E30">A19</font></span></a>, <a class="ref" href="rlvect_1.html#D10" onmouseover="rs('rlvect_1/D10')" onmouseout="rh()">RLVECT_1:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" onmouseover="rs('rlvect_1/T16')" onmouseout="rh()">RLVECT_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">jj</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c15">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">y</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D11" onmouseover="rs('algstr_0/D11')" onmouseout="rh()">ALGSTR_0:def 11</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #)</span><br/>

<span class="kw">thus </span><a NAME="E6:18_1_2_7"/>
<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> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E3:18_1_2_7"><span class="lab"><font color="Green" title="E30">A19</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:18_1_2"/><span class="lab"><font color="Green" title="E30">A20</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_2_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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/>

<span class="kw">reconsider </span><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="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">y</font> = 
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>
<br/>.= 
<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>

<br/>.= 
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:18_1_2"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_2_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="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/>

<span class="kw">reconsider </span><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="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="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> = 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">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"><span class="p2">(<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></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>
<br/>.= 
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>
<br/>.= 
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">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="c16">x1</font>,<span class="p2">(<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><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>
<br/>.= 
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:18_1_2"/>
(  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18_1_2"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="txt" href="#E4:18_1_2"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E7:18_1_2"><span class="lab"><font color="Green" title="E29">A18</font></span></a>, <a class="txt" href="#E3:18_1_2"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E6:18_1_2"><span class="lab"><font color="Green" title="E28">A17</font></span></a>, <a class="txt" href="#E2:18_1_2"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E1:18_1_2"><span class="lab"><font color="Green" title="E24">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:18_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</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="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E16">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:18_1"/>
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E4:18_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="rusub_1.html#D1" target="_self" onmouseover="rs('rusub_1/D1')" onmouseout="rh()">Def1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:18"/>
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
