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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">V</font> is  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a>  iff  <a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> is  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">O</font> =  <a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font>;<br/>
<span class="kw">thus </span><a NAME="E1:5"/>
( <font color="Maroon" title="c1">V</font> is  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a>  implies  <a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> is  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> is  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a>  implies <font color="Maroon" title="c1">V</font> is  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:5"/>
 <a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> is  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">V</font> is  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">A</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 <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"> such that </span><br/><a NAME="E4:5"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c3">A</font> is    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of  <a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_5.html#D1" onmouseover="rs('rlvect_5/D1')" onmouseout="rh()">RLVECT_5:def 1</a></span>;<br/>
<a NAME="E5:5"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 <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">addF</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> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#D4" onmouseover="rs('rlsub_1/D4')" onmouseout="rh()">RLSUB_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">A</font> as   <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> ;<br/>
<a NAME="E7:5"/>
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="rlvect_3.html#D3" onmouseover="rs('rlvect_3/D3')" onmouseout="rh()">RLVECT_3:def 3</a></span>;<br/>
<a NAME="E8:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A3</font></span>: 
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_5.html#T20" onmouseover="rs('rlvect_5/T20')" onmouseout="rh()">RLVECT_5:20</a></span>;<br/>
<a NAME="E9:5"/>
<font color="Maroon" title="c3">A</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="#E4:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="rlvect_3.html#D3" onmouseover="rs('rlvect_3/D3')" onmouseout="rh()">RLVECT_3:def 3</a></span>;<br/>
<a NAME="E10:5"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</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="ref" href="rlvect_5.html#T14" onmouseover="rs('rlvect_5/T14')" onmouseout="rh()">RLVECT_5:14</a></span>;<br/>
<a NAME="E11:5"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</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="txt" href="#E5:5"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E8:5"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="rlvect_3.html#D3" onmouseover="rs('rlvect_3/D3')" onmouseout="rh()">RLVECT_3:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:5"/>
<font color="Maroon" title="c1">V</font> is  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_5.html#D1" onmouseover="rs('rlvect_5/D1')" onmouseout="rh()">RLVECT_5:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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