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

<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   <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="Olive">b<sub>1</sub></font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>  <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>  holds <br/>( <font color="Olive" title="b2">K</font> is <font color="Maroon" title="c1">F</font> <a href="field_5.html#NV1" title="FIELD_5:NV.1">-infinite</a>  or ( <font color="Olive" title="b1">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a>  &amp;  <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>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</font> be   <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="c2">E</font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>  <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>  holds <br/>( <font color="Olive" title="b1">K</font> is <font color="Maroon" title="c1">F</font> <a href="field_5.html#NV1" title="FIELD_5:NV.1">-infinite</a>  or ( <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a>  &amp;  <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>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font> be  <font color="Maroon" title="c2">E</font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</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_5.html#NV1" title="FIELD_5:NV.1">-infinite</a>  or ( <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a>  &amp;  <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>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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>) ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">VK</font> =  <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>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">VE</font> =  <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>);<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">now </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"> ( <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>  implies ( <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a>  &amp;  <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>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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>) ) )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_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> 
 ; <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="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a>  &amp;  <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>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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>) )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">VK</font> =  <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>) as   <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c1">F</font> <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>;<br/><a NAME="E3:17_1"/><span class="lab"><font color="Green" title="E11">A</font></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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">VK</font>
 <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 NAME="E4:17_1"/><span class="lab"><font color="Green" title="E12">B</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="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">VK</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#T14" target="_self" onmouseover="rs('field_5/T14')" onmouseout="rh()">sp</a></span>;<br/><a NAME="E5:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">C</font></span>: 
 <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="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">VK</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T25" onmouseover="rs('vectsp_9/T25')" onmouseout="rh()">VECTSP_9:25</a></span>;<br/><span class="kw">thus </span><a NAME="E6:17_1"/>
<font color="Maroon" title="c2">E</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="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E12">B</font></span></a>, <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">  <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>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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>)</span><br/><span class="kw">thus </span><a NAME="E7:17_1"/>
 <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>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E11">A</font></span></a>, <a class="txt" href="#E5:17_1"><span class="lab"><font color="Green" title="E13">C</font></span></a>, <a class="ref" href="field_4.html#D7" onmouseover="rs('field_4/D7')" onmouseout="rh()">FIELD_4:def 7</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:17"/>
( <font color="Maroon" title="c3">K</font> is <font color="Maroon" title="c1">F</font> <a href="field_5.html#NV1" title="FIELD_5:NV.1">-infinite</a>  or ( <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a>  &amp;  <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>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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 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/>


</div>
