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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">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">V</font> be  <a href="matrlin.html#V2">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">GF</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>









<b>assume </b><a NAME="E1:46"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="vectsp_9.html#K2" target="_self">Subspaces_of</a> <font color="Maroon">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon">W1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E3:46"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:46"/><i><font color="Green">E40</font></i>: 
 <a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">W1</font> as  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T34">VECTSP_4:34</a></i>;<br/>
<a NAME="E6:46"/>
<font color="Maroon">W1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="vectsp_9.html#K2" target="_self">Subspaces_of</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E7:46"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="vectsp_9.html#K2" target="_self">Subspaces_of</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>


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