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<span class="kw">let </span><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#L1" title="RLVECT_1:struct.1">RLSStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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><br/> for <font color="Olive" title="b2">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b4">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="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> #)  st <font color="Olive" title="b4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> holds <br/><font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<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> #)  st <font color="Olive" title="b3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">w</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> <font color="Olive" title="b2">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<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> #)  st <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<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> #)  st <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<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> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> implies <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:8"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font></span><br/>

<a NAME="E2:8"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:8"/>
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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