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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/>

<b>reconsider </b><font color="Maroon">V'</font> =  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">V</font> as  <a href="rlvect_1.html#V2">strict</a> <a href="rlvect_1.html#NM2">RealLinearSpace</a> ;<br/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V'</font> = 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="rlsub_1.html#D3">RLSUB_1:def 3</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V'</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="rlsub_1.html#T19">RLSUB_1:19</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">V'</font></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_1.html#D3">RLSUB_1:def 3</a></i>
;<br/>
<a NAME="E3:29"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">V'</font> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">V'</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T40">RLSUB_1:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon">I</font> =  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V'</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V'</font> ;<br/>
<a NAME="E5:29"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">I</font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="rlvect_3.html#T8">RLVECT_3:8</a></i>;<br/>
<a NAME="E6:29"/>
 <a href="rlvect_3.html#K1">Lin</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">V'</font>
 
<b>by </b><i><a class="ref" href="rlvect_3.html#T19">RLVECT_3:19</a></i>;<br/>
<a NAME="E7:29"/><b>then </b>
<font color="Maroon">I</font> is    <a href="rlvect_3.html#M1">Basis</a> of <font color="Maroon">V'</font>
 
<b>by </b><i>, , <a class="ref" href="rlvect_3.html#D3">RLVECT_3:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:29"/>
 <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">V</font> is <a href="rlvect_5.html#V1" target="_self">finite-dimensional</a>
 <b>by </b><i/>;<br/>


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