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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a>  <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">  <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <span class="p1">(<span class="default"><a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">I</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E2:38"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c2">I</font> is    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_5.html#D1" target="_self" title="RLVECT_5:def.1">Def1</a></span>;<br/>
<span class="lab"><font color="Green" title="E34">A2</font></span>:  <a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> = 
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U5</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> #)
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#D4" title="RLSUB_1:def.4">RLSUB_1:def 4</a></span>
<br/>.= 
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c2">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="rlvect_3.html#D3" title="RLVECT_3:def.3">RLVECT_3:def 3</a></span>
;<br/>
<a NAME="E4:38"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c2">I</font> is  <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="rlvect_5.html#D3" target="_self" title="RLVECT_5:def.3">Def3</a>, <a class="ref" href="rlvect_3.html#D3" title="RLVECT_3:def.3">RLVECT_3:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:38"/>
 <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <span class="p1">(<span class="default"><a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="rlvect_5.html#T30" target="_self" title="RLVECT_5:th.30">Th30</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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