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<span class="kw">let </span><font color="Maroon" title="c3">K</font> be  <font color="Maroon" title="c1">F</font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>  <font color="Maroon" title="c2">E</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a>  <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">K</font> is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">BE</font> =  the   <a href="vectsp_7.html#NM1" title="VECTSP_7:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">BK</font> =  the   <a href="vectsp_7.html#NM1" title="VECTSP_7:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>;<br/>
<a NAME="E1:69_1_1"/>
 <a href="field_7.html#K7" title="FIELD_7:func.7">Base</a> ( the   <a href="vectsp_7.html#NM1" title="VECTSP_7:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>, the   <a href="vectsp_7.html#NM1" title="VECTSP_7:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>) is   <a href="vectsp_7.html#NM1" title="VECTSP_7:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#T29" target="_self" onmouseover="rs('field_7/T29')" onmouseout="rh()">BasKEF</a></span>;<br/>
<a NAME="E2:69_1_1"/><span class="kw">then </span>
 <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>) is  <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D1" onmouseover="rs('matrlin/D1')" onmouseout="rh()">MATRLIN:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:69_1_1"/>
<font color="Maroon" title="c3">K</font> is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D8" onmouseover="rs('field_4/D8')" onmouseout="rh()">FIELD_4:def 8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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