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<b>let </b><font color="Maroon">V</font> be  <a href="rusub_4.html#V1" target="_self">finite-dimensional</a> <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 ;<br/>

<b>set </b><font color="Maroon">W</font> =  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:18_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T2">RUSUB_3:2</a></i>;<br/><b>hence </b><a NAME="E3:18_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:18"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as  <a href="rlvect_3.html#V1">linearly-independent</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i>, <a class="ref" href="rusub_3.html#T23">RUSUB_3:23</a></i>;<br/>
<a NAME="E5:18"/>
 <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="rusub_3.html#T28">RUSUB_3:28</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as    <a href="rusub_3.html#M1">Basis</a> of  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/>
<a NAME="E7:18"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="rusub_4.html#K1" target="_self">dim</a> <span class="p1">(<span class="default"><a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:18"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="rusub_4.html#K1" target="_self">dim</a> <span class="p1">(<span class="default"><a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>


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