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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">W</font> be    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>







<b>assume </b><a NAME="E1:31"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="rusub_4.html#K2" target="_self">Subspaces_of</a> <font color="Maroon">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon">W1</font> being  <a href="bhsp_1.html#V1">strict</a>  <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E3:31"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:31"/><i><font color="Green">E27</font></i>: 
 <a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">W1</font> as  <a href="bhsp_1.html#V1">strict</a>  <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rusub_1.html#T21">RUSUB_1:21</a></i>;<br/>
<a NAME="E6:31"/>
<font color="Maroon">W1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="rusub_4.html#K2" target="_self">Subspaces_of</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="rusub_4.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E7:31"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="rusub_4.html#K2" target="_self">Subspaces_of</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>


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