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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">V</font> be  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a> <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">W</font> be    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>







<b>assume </b><a NAME="E1:49"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="rlvect_5.html#K2" title="RLVECT_5:func.2">Subspaces_of</a> <font color="Maroon" title="c3">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">W1</font> being  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c3">W</font><b> such that </b><br/><a NAME="E3:49"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c5">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>and </b><br/><a NAME="E4:49"/><i><font color="Green" title="E38">A2</font></i>: 
 <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c5">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="rlvect_5.html#D4" target="_self" title="RLVECT_5:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">W1</font> = <font color="Maroon" title="c5">W1</font> as  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="rlsub_1.html#T35" title="RLSUB_1:th.35">RLSUB_1:35</a></i>;<br/>
<a NAME="E6:49"/>
<font color="Maroon" title="c6">W1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="rlvect_5.html#K2" title="RLVECT_5:func.2">Subspaces_of</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="txt" href="rlvect_5.html#E4:49"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="rlvect_5.html#D4" target="_self" title="RLVECT_5:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E7:49"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="rlvect_5.html#K2" title="RLVECT_5:func.2">Subspaces_of</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="rlvect_5.html#E3:49"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>


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