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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>;<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>reconsider </b><font color="Maroon" title="c3">V'</font> = <font color="Maroon" title="c1">V</font> as   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a> ;<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="c3">V'</font>;<br/>
<a NAME="E3:17"/><i><font color="Green" title="E13">A1</font></i>: 
 <a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V'</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c3">V'</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c3">V'</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c3">V'</font>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c3">V'</font> #)
 
<b>by </b><i><a class="ref" href="rusub_3.html#D2" title="RUSUB_3:def.2">RUSUB_3:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">I</font> = <font color="Maroon" title="c4">I</font> as  <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>by </b><i><a class="ref" href="rusub_4.html#T3" target="_self" title="RUSUB_4:th.3">Th3</a></i>;<br/>
<a NAME="E5:17"/><i><font color="Green" title="E14">A2</font></i>: 
 <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">I</font>
 
<b>by </b><i><a class="ref" href="rusub_4.html#D3" target="_self" title="RUSUB_4:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">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>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">W</font> ;<br/>
<a NAME="E8:17"/>
<font color="Maroon" title="c7">A</font> is <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="rusub_3.html#D2" title="RUSUB_3:def.2">RUSUB_3:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c7">A</font> as  <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a> <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="rusub_3.html#T22" title="RUSUB_3:th.22">RUSUB_3:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">A'</font> = <font color="Maroon" title="c8">B</font> as  <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>by </b><i><a class="ref" href="rusub_4.html#T3" target="_self" title="RUSUB_4:th.3">Th3</a></i>;<br/>
<a NAME="E11:17"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">A'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E3:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="rusub_4.html#T2" target="_self" title="RUSUB_4:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E12:17"/>
 <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Maroon" title="c2">W</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>
 <b>by </b><i><a class="txt" href="rusub_4.html#E5:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="rusub_4.html#D3" target="_self" title="RUSUB_4:def.3">Def3</a></i>;<br/>


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