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<b>let </b><font color="Maroon" title="c1">GF</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">VectSp</a> of <font color="Maroon" title="c1">GF</font>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c2">V</font> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c2">V</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">A'</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">B'</font> = <font color="Maroon" title="c4">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="c2">V</font> <b>by </b><i><a class="txt" href="vectsp_9.html#E1:25"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="vectsp_9.html#T24" target="_self" title="VECTSP_9:th.24">Th24</a></i>;<br/>
<a NAME="E3:25"/><i><font color="Green" title="E25">A2</font></i>: 
 <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <font color="Maroon" title="c2">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E4:25"/>
<font color="Maroon" title="c6">B'</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E5:25"/><b>then </b><i><font color="Green" title="E26">A3</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">B'</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">A'</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E3:25"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="vectsp_9.html#T23" target="_self" title="VECTSP_9:th.23">Th23</a></i>;<br/>
<a NAME="E6:25"/><i><font color="Green" title="E27">A4</font></i>: 
 <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <font color="Maroon" title="c2">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E7:25"/>
<font color="Maroon" title="c5">A'</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E8:25"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">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="c6">B'</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E6:25"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="vectsp_9.html#T23" target="_self" title="VECTSP_9:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E9:25"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">A</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="c4">B</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E5:25"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


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