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<b>let </b><font color="Maroon">GF</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">GF</font>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">V</font> is <a href="matrlin.html#V2">finite-dimensional</a>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">V</font>, <font color="Maroon">B</font> be    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">V</font>;<br/>

<b>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>, </i>;<br/>
<a NAME="E3:25"/><i><font color="Green">E40</font></i>: 
 <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</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="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E4:25"/>
<font color="Maroon">B'</font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E5:25"/><b>then </b><i><font color="Green">E42</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="E6:25"/><i><font color="Green">E43</font></i>: 
 <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</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="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E7:25"/>
<font color="Maroon">A'</font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E8:25"/><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="E9:25"/>
 <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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