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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">V1</font> is <a href="rlsub_1.html#V1" title="RLSUB_1:attr.1">linearly-closed</a> &amp; not <font color="Olive" title="b1">V1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 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="rsspace.html#K10" title="RSSPACE:func.10">Zero_</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="rsspace.html#K8" title="RSSPACE:func.8">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="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <font color="Olive" title="b1">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span> #) is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font></span><br/><b>let </b><font color="Maroon" title="c2">V1</font> be   <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"> ( <font color="Maroon" title="c2">V1</font> is <a href="rlsub_1.html#V1" title="RLSUB_1:attr.1">linearly-closed</a> &amp; not <font color="Maroon" title="c2">V1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 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="rsspace.html#K10" title="RSSPACE:func.10">Zero_</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="rsspace.html#K8" title="RSSPACE:func.8">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="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span> #) is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> )</span><br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c2">V1</font> is <a href="rlsub_1.html#V1" title="RLSUB_1:attr.1">linearly-closed</a> &amp; not <font color="Maroon" title="c2">V1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="rsspace.html#K10" title="RSSPACE:func.10">Zero_</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="rsspace.html#K8" title="RSSPACE:func.8">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="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span> #) is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font></span><br/>

<a NAME="E2:25"/><i><font color="Green" title="E23">A2</font></i>: 
 <a href="rsspace.html#K10" title="RSSPACE:func.10">Zero_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">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="c1">V</font>
 
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="rsspace.html#D10" target="_self" title="RSSPACE:def.10">Def10</a></i>;<br/>
<a NAME="E3:25"/><i><font color="Green" title="E24">A3</font></i>: 
 <a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">V1</font>
 
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="rsspace.html#D8" target="_self" title="RSSPACE:def.8">Def8</a></i>;<br/>
<a NAME="E4:25"/>
 <a href="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</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="txt" href="#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="rsspace.html#D9" target="_self" title="RSSPACE:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E5:25"/>
 <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="rsspace.html#K10" title="RSSPACE:func.10">Zero_</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="rsspace.html#K8" title="RSSPACE:func.8">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="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span> #) is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E2:25"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E3:25"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="rlsub_1.html#T32" title="RLSUB_1:th.32">RLSUB_1:32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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