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<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">RLSStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V1</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="lpspace1.html#V1" title="LPSPACE1:attr.1">multi-closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V1</font> holds <br/>(  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Olive" title="b1">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Olive" title="b1">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Olive" title="b1">V1</font></span>)</span> #) is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Olive" title="b1">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Olive" title="b1">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Olive" title="b1">V1</font></span>)</span> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Olive" title="b1">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Olive" title="b1">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Olive" title="b1">V1</font></span>)</span> #) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Olive" title="b1">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Olive" title="b1">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Olive" title="b1">V1</font></span>)</span> #) is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> )</span><br/><b>let </b><font color="Maroon" title="c2">V1</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="lpspace1.html#V1" title="LPSPACE1:attr.1">multi-closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V1</font> implies (  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Maroon" title="c2">V1</font></span>)</span> #) is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Maroon" title="c2">V1</font></span>)</span> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Maroon" title="c2">V1</font></span>)</span> #) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Maroon" title="c2">V1</font></span>)</span> #) is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> ) )</span><br/>



<b>assume </b><a NAME="E1:6"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Maroon" title="c2">V1</font></span>)</span> #) is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Maroon" title="c2">V1</font></span>)</span> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Maroon" title="c2">V1</font></span>)</span> #) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Maroon" title="c2">V1</font></span>)</span> #) is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> )</span><br/>

<a NAME="E2:6"/><b>then </b>
(  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> &amp;  <a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6"/>
(  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Maroon" title="c2">V1</font></span>)</span> #) is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Maroon" title="c2">V1</font></span>)</span> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Maroon" title="c2">V1</font></span>)</span> #) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">V1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K1" title="LPSPACE1:func.1">Mult_</a> <font color="Maroon" title="c2">V1</font></span>)</span> #) is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> )
 <b>by </b><i><a class="ref" href="lpspace1.html#T2" target="_self" title="LPSPACE1:th.2">RLSUB132</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
