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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">E1</font>, <font color="Olive" title="b2">E2</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>  st <font color="Olive" title="b1">E1</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a> <font color="Olive" title="b2">E2</font> holds <br/> <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Olive" title="b1">E1</font>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Olive" title="b2">E2</font>,<font color="Maroon" title="c1">F</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">E1</font>, <font color="Maroon" title="c3">E2</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"> ( <font color="Maroon" title="c2">E1</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a> <font color="Maroon" title="c3">E2</font> implies  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c3">E2</font>,<font color="Maroon" title="c1">F</font>) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">V1</font> =  <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c1">F</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">V2</font> =  <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">E2</font>,<font color="Maroon" title="c1">F</font>);<br/>
<span class="kw">assume </span><a NAME="E1:7"/>
<font color="Maroon" title="c2">E1</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a> <font color="Maroon" title="c3">E2</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="c2">E1</font>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c3">E2</font>,<font color="Maroon" title="c1">F</font>)</span><br/>

<a NAME="E2:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">AS</font></span>: 
 <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">E2</font>,<font color="Maroon" title="c1">F</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#T4" target="_self" onmouseover="rs('field_7/T4')" onmouseout="rh()">str12</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:7_1"/>
(  <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c1">F</font>) is  <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  or  not  <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c1">F</font>) is  <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><span class="kw">suppose </span></a><a NAME="E1:7_1_1"/><span class="lab"><font color="Green" title="E6">F</font></span>: 
 <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c1">F</font>) is  <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</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">  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c3">E2</font>,<font color="Maroon" title="c1">F</font>)</span><br/><div class="add"><span class="kw">hence </span> <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E1</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">E2</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E5">AS</font></span></a>, <a class="ref" href="field_4.html#D7" onmouseover="rs('field_4/D7')" onmouseout="rh()">FIELD_4:def 7</a></span>
<br/>.= 
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c3">E2</font>,<font color="Maroon" title="c1">F</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E5">AS</font></span></a>, <a class="txt" href="#E1:7_1_1"><span class="lab"><font color="Green" title="E6">F</font></span></a>, <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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2"><span class="kw">suppose </span></a><a NAME="E1:7_1_2"/><span class="lab"><font color="Green" title="E6">F</font></span>: 
 not  <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c1">F</font>) is  <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</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">  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c3">E2</font>,<font color="Maroon" title="c1">F</font>)</span><br/><div class="add"><span class="kw">hence </span> <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c1">F</font>) = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
<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="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c3">E2</font>,<font color="Maroon" title="c1">F</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E5">AS</font></span></a>, <a class="txt" href="#E1:7_1_2"><span class="lab"><font color="Green" title="E6">F</font></span></a>, <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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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