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<b>let </b><font color="Maroon" title="c1">GF</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">GF</font>;<br/><b>let </b><font color="Maroon" title="c3">W</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c4">A</font> be    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c3">W</font>;<br/>







<a NAME="E1:15"/>
( <font color="Maroon" title="c4">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c4">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/>
<b>then reconsider </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c4">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 <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_9.html#T15" target="_self" title="VECTSP_9:th.15">Th15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">I</font> being    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E4:15"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">I</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T27" title="VECTSP_7:th.27">VECTSP_7:27</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">I</font>
;<br/>

<b>thus </b><a NAME="E5:15"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">I</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E4:15"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>


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