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<b>consider </b><font color="Maroon">ZS</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">O</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">ZS</font>, <font color="Maroon">F</font> being   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">ZS</font>, <font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">ZS</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">ZS</font><b> such that </b><br/><a NAME="E2:15_1_1"/><i><font color="Green">E30</font></i>: 
( ( for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) holds  <font color="Olive">v</font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Olive">v</font> ) &amp; ( for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) holds  <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Olive">w</font></span>)</span> ) &amp; ( for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) holds  <font color="Olive">v</font> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" target="_self">0.</a> <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> ) &amp; ( for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #)  ex <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) st <font color="Olive">v</font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1" target="_self">0.</a> <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) ) &amp; ( for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) holds  <font color="Olive">a</font> <a href="rlvect_1.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Olive">w</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3" target="_self">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3" target="_self">*</a> <font color="Olive">w</font></span>)</span> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K3">+</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3" target="_self">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3" target="_self">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3" target="_self">*</a> <font color="Olive">v</font></span>)</span> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3" target="_self">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="rlvect_1.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3" target="_self">*</a> <font color="Olive">v</font></span>)</span> ) &amp; ( for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) holds  1 <a href="rlvect_1.html#K3" target="_self">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> ) )
 <b>by </b><i/>;<br/>
<b>take </b>
 <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #)
;<br/>

<b>thus </b><a NAME="E3:15_1_1"/>
not  <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) is <a href="struct_0.html#V3">empty</a>
 ;<br/>

<b>thus </b><a NAME="E4:15_1_1"/>
(  <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) is <a href="rlvect_1.html#V2" target="_self">strict</a> &amp;  <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) is <a href="rlvect_1.html#V3" target="_self">Abelian</a> &amp;  <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) is <a href="rlvect_1.html#V4" target="_self">add-associative</a> &amp;  <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) is <a href="rlvect_1.html#V5" target="_self">right_zeroed</a> &amp;  <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) is <a href="rlvect_1.html#V6" target="_self">right_complementable</a> &amp;  <a href="rlvect_1.html#G2" target="_self">RLSStruct</a>(# <font color="Maroon">ZS</font>,<font color="Maroon">O</font>,<font color="Maroon">F</font>,<font color="Maroon">G</font> #) is <a href="rlvect_1.html#V7" target="_self">RealLinearSpace-like</a> )
 <b>by </b><i>, , , , , </i>;<br/>


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