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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="rusub_4.html#V1" target="_self">finite-dimensional</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:12"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T24">RUSUB_3:24</a></i>;<br/>
<a NAME="E5:12"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E1:12"><i><font color="Green">E11</font></i></a>, <a class="ref" href="rusub_4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:12"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E4:12"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>hence </b><a NAME="E7:12"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="rusub_4.html#V1" target="_self">finite-dimensional</a>
 <b>by </b><i><a class="ref" href="rusub_4.html#D1" target="_self">Def1</a></i>;<br/>


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