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<span class="kw">let </span><font color="Maroon" title="c1">r</font>, <font color="Maroon" title="c2">s</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">R</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>   <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>   <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>   <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>   <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">RLSStruct</a> <br/> for <font color="Olive" title="b2">LR</font> being   <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Olive" title="b1">R</font> holds  <font color="Maroon" title="c1">r</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Olive" title="b2">LR</font></span>)</span> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Olive" title="b2">LR</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">R</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>   <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>   <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>   <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</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">LR</font> being   <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c3">R</font> holds  <font color="Maroon" title="c1">r</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Olive" title="b1">LR</font></span>)</span> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Olive" title="b1">LR</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">LR</font> be    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c3">R</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="c1">r</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font></span>)</span> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font></span><br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:34_1"/>
( <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or ( <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_1_1"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
( <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <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="c1">r</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font></span>)</span> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font></span><br/><div class="add"><a NAME="E2:34_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a>  <a href="rlvect_2.html#K4" title="RLVECT_2:func.4">ZeroLC</a> <font color="Maroon" title="c3">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#D2" target="_self" title="RLAFFIN1:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:34_1_1"/>
<font color="Maroon" title="c1">r</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font></span>)</span> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="rlaffin1.html#D2" target="_self" title="RLAFFIN1:def.2">Def2</a>, <a class="ref" href="rlaffin1.html#T26" target="_self" title="RLAFFIN1:th.26">Th26</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_1_2"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
( <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <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="c1">r</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font></span>)</span> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_2"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="rlaffin1.html#D2" target="_self" title="RLAFFIN1:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c4">LR</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_2"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="rlaffin1.html#D2" target="_self" title="RLAFFIN1:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c4">LR</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
<font color="Maroon" title="c4">LR</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T205" title="XCMPLX_1:th.205">XCMPLX_1:205</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_2"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="rlaffin1.html#D2" target="_self" title="RLAFFIN1:def.2">Def2</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/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:34_1_2"/>
<font color="Maroon" title="c1">r</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font></span>)</span> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="rlaffin1.html#K2" title="RLAFFIN1:func.2">(*)</a> <font color="Maroon" title="c4">LR</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D11" title="RLVECT_2:def.11">RLVECT_2:def 11</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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