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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">W</font> be    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c1">V</font> is <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">A</font> being    <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Maroon" title="c2">W</font>;<br/>
<b>consider </b><font color="Maroon" title="c4">I</font> being    <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:12"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T24" title="RUSUB_3:th.24">RUSUB_3:24</a></i>;<br/>
<a NAME="E5:12"/>
<font color="Maroon" title="c4">I</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E1:12"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="rusub_4.html#T3" target="_self" title="RUSUB_4:th.3">Th3</a></i>;<br/>
<a NAME="E6:12"/><b>then </b>
<font color="Maroon" title="c3">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E4:12"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<b>hence </b><a NAME="E7:12"/>
<font color="Maroon" title="c2">W</font> is <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a>
 <b>by </b><i><a class="ref" href="rusub_4.html#D1" target="_self" title="RUSUB_4:def.1">Def1</a></i>;<br/>


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