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<b>consider </b><font color="Maroon">A</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E2:31_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">A</font> is    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">V</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:31_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>
 ;<br/>
<a NAME="E5:31_1_1"/>
for <font color="Olive">I</font> being   <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">V</font> holds   <a href="card_1.html#K1">Card</a> <font color="Olive">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E6:31_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">I</font> being   <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">V</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">I</font>
 ;<br/>


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