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

<b>let </b><font color="Maroon" title="c1">GF</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">GF</font><br/> for <font color="Olive" title="b2">W</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b1">V</font> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> holds <br/><font color="Olive" title="b2">W</font> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c2">V</font> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> holds <br/><font color="Olive" title="b1">W</font> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">V</font> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> implies <font color="Maroon" title="c3">W</font> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> )</span><br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c2">V</font> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">W</font> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a></span><br/>

<b>consider </b><font color="Maroon" title="c4">A</font> being    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c3">W</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">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="E4:27"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">I</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T17" target="_self" title="VECTSP_9:th.17">Th17</a></i>;<br/>
<a NAME="E5:27"/>
<font color="Maroon" title="c5">I</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E1:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="vectsp_9.html#T24" target="_self" title="VECTSP_9:th.24">Th24</a></i>;<br/>
<a NAME="E6:27"/><b>then </b>
<font color="Maroon" title="c4">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E4:27"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<b>hence </b><a NAME="E7:27"/>
<font color="Maroon" title="c3">W</font> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a>
 <b>by </b><i><a class="ref" href="matrlin.html#D3" title="MATRLIN:def.3">MATRLIN:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
