<?xml version="1.0"?>
<div class="add">

<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="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">VectSp</a> of <font color="Maroon" title="c1">GF</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>;<br/>





<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c3">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">W</font> =  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/><a NAME="E2:33_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</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#T13" title="VECTSP_7:th.13">VECTSP_7:13</a></i>;<br/><b>hence </b><a NAME="E3:33_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:33"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c3">A</font> as  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <b>by </b><i><a class="txt" href="vectsp_9.html#E1:33"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="vectsp_9.html#T16" target="_self" title="VECTSP_9:th.16">Th16</a></i>;<br/>
<a NAME="E5:33"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</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="c5">B</font>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T21" target="_self" title="VECTSP_9:th.21">Th21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c5">B</font> as    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of  <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="E7:33"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#D2" target="_self" title="VECTSP_9:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E8:33"/>
 <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="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span>
 ;<br/>


</div>
