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<b>let </b><font color="Maroon" title="c1">V</font> be  <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a> <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Maroon" title="c1">V</font> holds <br/> ex <font color="Olive" title="b2">W</font> being <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st  <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font></span><br/><b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Maroon" title="c1">V</font> implies  ex <font color="Olive" title="b1">W</font> being <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st  <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> )</span><br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Maroon" title="c1">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st  <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font></span><br/>

<b>consider </b><font color="Maroon" title="c3">I</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c3">I</font> is    <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rusub_4.html#D2" target="_self" title="RUSUB_4:def.2">Def2</a></i>;<br/>
<a NAME="E4:27"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">I</font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E1:27"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="rusub_4.html#E3:27"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="rusub_4.html#D3" target="_self" title="RUSUB_4:def.3">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">A</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">I</font><b> such that </b><br/><a NAME="E6:27"/><i><font color="Green" title="E20">A3</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T87" title="FINSEQ_4:th.87">FINSEQ_4:87</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">W</font> =  <a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c5">A</font> as  <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a> <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E9:27"/>
<font color="Maroon" title="c3">I</font> is <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E3:27"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="rusub_3.html#D2" title="RUSUB_3:def.2">RUSUB_3:def 2</a></i>;<br/>
<a NAME="E10:27"/><b>then </b>
 <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E6:27"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="rusub_4.html#T9" target="_self" title="RUSUB_4:th.9">Th9</a>, <a class="ref" href="rlvect_3.html#T6" title="RLVECT_3:th.6">RLVECT_3:6</a></i>;<br/>
<b>hence </b><a NAME="E11:27"/>
 ex <font color="Olive" title="b1">W</font> being <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st  <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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