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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">W</font> being   <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>  st <font color="Maroon" title="c1">V</font> is  <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a>  holds <br/><font color="Olive" title="b1">W</font> is  <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">V</font> is  <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a>  implies <font color="Maroon" title="c2">W</font> is  <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a>  )</span><br/>



<span class="kw">consider </span><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/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:11"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_3.html#T24" title="RUSUB_3:th.24">RUSUB_3:24</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:11"/>
<font color="Maroon" title="c1">V</font> is  <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">W</font> is  <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a> </span><br/>

<a NAME="E5:11"/><span class="kw">then </span>
<font color="Maroon" title="c4">I</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_4.html#T3" target="_self" title="RUSUB_4:th.3">Th3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:11"/>
<font color="Maroon" title="c2">W</font> is  <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="rusub_4.html#D1" target="_self" title="RUSUB_4:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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