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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L1</font>, <font color="Olive" title="b2">L2</font> being   <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font> holds  <span class="p1">(<span class="default"><a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">L1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b2">L2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L1</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Olive" title="b2">L2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">L1</font>, <font color="Maroon" title="c3">L2</font> be    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</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"> <span class="p1">(<span class="default"><a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c3">L2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L1</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c3">L2</font></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">v2</font> =  <a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c3">L2</font>;<br/>
<a NAME="E1:109"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
( <font color="Maroon" title="c2">L1</font> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a>  <a href="rlvect_2.html#K12" title="RLVECT_2:func.12">@</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K13" title="RLVECT_2:func.13">@</a> <font color="Maroon" title="c2">L1</font></span>)</span> &amp; <font color="Maroon" title="c3">L2</font> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a>  <a href="rlvect_2.html#K12" title="RLVECT_2:func.12">@</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K13" title="RLVECT_2:func.13">@</a> <font color="Maroon" title="c3">L2</font></span>)</span> )
 
;<br/>
<a NAME="E2:109"/>
<font color="Maroon" title="c3">L2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D16" target="_self" title="RLVECT_2:def.16">Def16</a></span>;<br/>
<a NAME="E3:109"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c3">L2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E4:109"/>
<font color="Maroon" title="c2">L1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D16" target="_self" title="RLVECT_2:def.16">Def16</a></span>;<br/>
<a NAME="E5:109"/><span class="kw">then </span>
<font color="Maroon" title="c2">L1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c3">L2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" title="RLVECT_2:func.14">LCAdd</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">L1</font>,<span class="p2">(<span class="default"><a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c3">L2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D1" target="_self" title="RLVECT_2:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" title="RLVECT_2:func.14">LCAdd</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K13" title="RLVECT_2:func.13">@</a> <font color="Maroon" title="c2">L1</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" title="RLVECT_2:func.13">@</a> <font color="Maroon" title="c3">L2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="rlvect_2.html#D1" target="_self" title="RLVECT_2:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c2">L1</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c3">L2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="rlvect_2.html#D19" target="_self" title="RLVECT_2:def.19">Def19</a></span>
;<br/> <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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