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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <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">  for <font color="Olive" title="b1">E</font> being  <font color="Maroon" title="c1">F</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><br/>  for <font color="Olive" title="b2">K</font> being  <font color="Maroon" title="c1">F</font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>  <font color="Olive">b<sub>1</sub></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> holds   <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Olive" title="b2">K</font>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Olive" title="b2">K</font>,<font color="Olive" title="b1">E</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Olive" title="b1">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</font> be  <font color="Maroon" title="c1">F</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">  for <font color="Olive" title="b1">K</font> being  <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> holds   <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Olive" title="b1">K</font>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><br/><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">  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></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="E1:67"/>
 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> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E2:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">B1</font></span>: 
 <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>) 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/>
<a NAME="E3:67"/>
 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="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E4:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">B2</font></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="c2">E</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/>
<a NAME="E5:67"/><span class="lab"><font color="Green" title="E40">A</font></span>: 
 <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="E6:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">B</font></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 href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>) = 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <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_4.html#D7" onmouseover="rs('field_4/D7')" onmouseout="rh()">FIELD_4:def 7</a></span>
<br/>.= 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><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>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#D1" onmouseover="rs('vectsp_9/D1')" onmouseout="rh()">VECTSP_9:def 1</a>, <a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E41">B</font></span></a>, <a class="txt" href="#E5:67"><span class="lab"><font color="Green" title="E40">A</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</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></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</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="c3">K</font>,<font color="Maroon" title="c2">E</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#T21" target="_self" onmouseover="rs('field_7/T21')" onmouseout="rh()">BE2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<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></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</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="c3">K</font>,<font color="Maroon" title="c2">E</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#D1" onmouseover="rs('vectsp_9/D1')" onmouseout="rh()">VECTSP_9:def 1</a>, <a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E38">B1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</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="c3">K</font>,<font color="Maroon" title="c2">E</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D7" onmouseover="rs('field_4/D7')" onmouseout="rh()">FIELD_4:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#D1" onmouseover="rs('vectsp_9/D1')" onmouseout="rh()">VECTSP_9:def 1</a>, <a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E39">B2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D7" onmouseover="rs('field_4/D7')" onmouseout="rh()">FIELD_4:def 7</a></span>
;<br/> <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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