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<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">V</font>, <font color="Maroon">B</font> be    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E1:10"/>
<font color="Maroon">V</font> is <a href="rusub_4.html#V1" target="_self">finite-dimensional</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font>, <font color="Maroon">B'</font> = <font color="Maroon">B</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<a NAME="E3:10"/><i><font color="Green">E26</font></i>: 
 <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/>
<a NAME="E4:10"/>
<font color="Maroon">B'</font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/>
<a NAME="E5:10"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">B'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:10"/><i><font color="Green">E29</font></i>: 
 <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/>
<a NAME="E7:10"/>
<font color="Maroon">A'</font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/>
<a NAME="E8:10"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">A'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">B'</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:10"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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