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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   <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">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a>  iff  ex <font color="Olive" title="b2">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">E</font> st <font color="Olive" title="b1">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b2">T</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"> ( <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>  iff  ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font> st <font color="Maroon" title="c2">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b1">T</font>) )</span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">F</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a><br/>  for <font color="Olive" title="b2">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Olive" title="b1">F</font>  st  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Olive" title="b2">E</font>,<font color="Olive" title="b1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> ex <font color="Olive" title="b3">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Olive">b<sub>1</sub></font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">E</font> st <font color="Olive" title="b2">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">T</font>);<br/>
<div><span class="lab"><font color="Green" title="E47">IA</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_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">  for <font color="Olive" title="b1">F</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a><br/>  for <font color="Olive" title="b2">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Olive" title="b1">F</font>  st  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Olive" title="b2">E</font>,<font color="Olive" title="b1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> ex <font color="Olive" title="b3">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Olive">b<sub>1</sub></font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">E</font> st <font color="Olive" title="b2">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">T</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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="c3">F</font>  st  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Olive" title="b1">E</font>,<font color="Maroon" title="c3">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> ex <font color="Olive" title="b2">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c3">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">E</font> st <font color="Olive" title="b1">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c3">F</font>,<font color="Olive" title="b2">T</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">E</font> be   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c3">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="c4">E</font>,<font color="Maroon" title="c3">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c3">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">E</font> st <font color="Maroon" title="c4">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c3">F</font>,<font color="Olive" title="b1">T</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:83_1"/>
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c3">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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">  ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c3">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">E</font> st <font color="Maroon" title="c4">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c3">F</font>,<font color="Olive" title="b1">T</font>)</span><br/><a NAME="E2:83_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a> <font color="Maroon" title="c4">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#T8" target="_self" onmouseover="rs('field_7/T8')" onmouseout="rh()">str1a</a></span>;<br/><a NAME="E3:83_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">E</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">T</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">E</font> ;<br/><a NAME="E5:83_1"/>
<font color="Maroon" title="c5">T</font> is <font color="Maroon" title="c3">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">T</font> = <font color="Maroon" title="c5">T</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c3">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">E</font> ;<br/><a NAME="E7:83_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">F</font>
 ;<br/><a NAME="E8:83_1"/><span class="kw">then </span>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c6">T</font>) <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#T3" target="_self" onmouseover="rs('field_7/T3')" onmouseout="rh()">Th1</a></span>;<br/><a NAME="E9:83_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c6">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_1"><span class="lab"><font color="Green" title="E48">A</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:83_1"/>
 ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c3">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">E</font> st <font color="Maroon" title="c4">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c3">F</font>,<font color="Olive" title="b1">T</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><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E48">IS</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_2"><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">  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:83_2"/><span class="lab"><font color="Green" title="E49">AS</font></span>: 
( <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</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">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_2_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">  for <font color="Olive" title="b1">F</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a><br/>  for <font color="Olive" title="b2">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Olive" title="b1">F</font>  st  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Olive" title="b2">E</font>,<font color="Olive" title="b1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> holds <br/> ex <font color="Olive" title="b3">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Olive">b<sub>1</sub></font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">E</font> st <font color="Olive" title="b2">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">T</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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="c4">F</font>  st  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Olive" title="b1">E</font>,<font color="Maroon" title="c4">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> holds <br/> ex <font color="Olive" title="b2">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Olive">b<sub>2</sub></font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive">b<sub>3</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b2">T</font>,<font color="Olive">b<sub>4</sub></font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">E</font> be   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c4">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="c5">E</font>,<font color="Maroon" title="c4">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> implies  ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Olive">b<sub>1</sub></font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">T</font>,<font color="Olive">b<sub>3</sub></font>) )</span><br/><span class="kw">assume </span><a NAME="E1:83_2_1"/><span class="lab"><font color="Green" title="E50">Z</font></span>: 
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c4">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</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">  ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Olive">b<sub>1</sub></font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">T</font>,<font color="Olive">b<sub>3</sub></font>)</span><br/><a NAME="E2:83_2_1"/><span class="kw">then </span>
 <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c4">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="field_4.html#D7" onmouseover="rs('field_4/D7')" onmouseout="rh()">FIELD_4:def 7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">K</font> = <font color="Maroon" title="c5">E</font> as  <font color="Maroon" title="c4">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="c4">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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_1_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="E1:83_2_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2"><span class="lab"><font color="Green" title="E49">AS</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:83_2_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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">  ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Olive">b<sub>1</sub></font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">T</font>,<font color="Olive">b<sub>3</sub></font>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:83_2_1_1_1"/>
 ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c4">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">E</font> st <font color="Maroon" title="c5">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<font color="Olive" title="b1">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_1"><span class="lab"><font color="Green" title="E50">Z</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E47">IA</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:83_2_1_1_2"/><span class="lab"><font color="Green" title="E51">B0</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <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">  ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Olive">b<sub>1</sub></font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">T</font>,<font color="Olive">b<sub>3</sub></font>)</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_2_1_1_2_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">  not <!--HUMANRECFAILED--> for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">K</font> holds  <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">F</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:83_2_1_1_2_1"/><span class="lab"><font color="Green" title="E52">B1</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">K</font> holds  <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">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"> contradiction</span><br/><a NAME="E2:83_2_1_1_2_1"/>
 for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">E</font> iff <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">F</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_2_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">E</font> iff <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">F</font> )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_2_1_1_2_1_1_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="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">F</font> implies <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">E</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:83_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E53">B</font></span>: 
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">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="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">E</font></span><br/><a NAME="E2:83_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c4">F</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c5">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T7" onmouseover="rs('field_4/T7')" onmouseout="rh()">FIELD_4:7</a></span>;<br/><a NAME="E3:83_2_1_1_2_1_1_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#D1" onmouseover="rs('ec_pf_1/D1')" onmouseout="rh()">EC_PF_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:83_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E53">B</font></span></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="E2:83_2_1_1_2_1_1"/>
( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">E</font> iff <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_1_1_2_1"><span class="lab"><font color="Green" title="E52">B1</font></span></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="E3:83_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_1"><span class="lab"><font color="Green" title="E50">Z</font></span></a>, <a class="txt" href="#E1:83_2_1_1_2"><span class="lab"><font color="Green" title="E51">B0</font></span></a>, <a class="ref" href="field_7.html#T7" target="_self" onmouseover="rs('field_7/T7')" onmouseout="rh()">quah1</a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</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">then </span><span class="kw">consider </span><font color="Maroon" title="c7">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">K</font><span class="kw"> such that </span><br/><a NAME="E4:83_2_1_1_2"/><span class="lab"><font color="Green" title="E52">C1</font></span>: 
 not <font color="Maroon" title="c7">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">F</font>
 ;<br/><span class="kw">set </span><font color="Maroon" title="c8">p</font> =  <a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c4">F</font>);<br/><a NAME="E5:83_2_1_1_2"/><span class="lab"><font color="Green" title="E53">C2</font></span>: 
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> is <font color="Maroon" title="c4">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a> 
 ;<br/><a NAME="E6:83_2_1_1_2"/><span class="lab"><font color="Green" title="E54">C3</font></span>: 
(  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">F</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp;  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">F</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_2_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:83_2_1_1_2_2"/>
<font color="Maroon" title="c6">K</font> is  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T7" onmouseover="rs('field_4/T7')" onmouseout="rh()">FIELD_4:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:83_2_1_1_2_2"/>
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">F</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_1"><span class="lab"><font color="Green" title="E50">Z</font></span></a>, <a class="ref" href="field_5.html#T15" onmouseover="rs('field_5/T15')" onmouseout="rh()">FIELD_5:15</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> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">F</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1</span><br/>

<a NAME="E3:83_2_1_1_2_2"/><span class="lab"><font color="Green" title="E55">C5</font></span>: 
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T35" onmouseover="rs('field_6/T35')" onmouseout="rh()">FIELD_6:35</a></span>;<br/>
<a NAME="E4:83_2_1_1_2_2"/>
<font color="Maroon" title="c7">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:83_2_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">C6</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_2_1_1_2_2"><span class="lab"><font color="Green" title="E55">C5</font></span></a></span>;<br/>
<a NAME="E6:83_2_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">F</font>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E7:83_2_1_1_2_2"/><span class="kw">then </span>
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">F</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:83_2_1_1_2_2"/>
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">F</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_2_1_1_2_2"><span class="lab"><font color="Green" title="E56">C6</font></span></a>, <a class="txt" href="#E4:83_2_1_1_2"><span class="lab"><font color="Green" title="E52">C1</font></span></a>, <a class="ref" href="field_7.html#T7" target="_self" onmouseover="rs('field_7/T7')" onmouseout="rh()">quah1</a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</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><a NAME="E7:83_2_1_1_2"/>
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p1">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c4">F</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">F</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T67" onmouseover="rs('field_6/T67')" onmouseout="rh()">FIELD_6:67</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_1_1_2_3"><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="E1:83_2_1_1_2_3"/><span class="kw">then </span>
(  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p1">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c4">F</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> or  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p1">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c4">F</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_2_1_1_2"><span class="lab"><font color="Green" title="E54">C3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_1_1_2_3_1"><span class="kw">suppose </span></a><a NAME="E1:83_2_1_1_2_3_1"/>
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p1">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c4">F</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</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">  ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Olive">b<sub>1</sub></font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">T</font>,<font color="Olive">b<sub>3</sub></font>)</span><br/><div class="add"><a NAME="E2:83_2_1_1_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">K</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_1"><span class="lab"><font color="Green" title="E50">Z</font></span></a>, <a class="ref" href="field_7.html#T36" target="_self" onmouseover="rs('field_7/T36')" onmouseout="rh()">ft1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:83_2_1_1_2_3_1"/>
 ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c4">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">E</font> st <font color="Maroon" title="c5">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<font color="Olive" title="b1">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_2_1_1_2"><span class="lab"><font color="Green" title="E53">C2</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_1_1_2_3_2"><span class="kw">suppose </span></a><a NAME="E1:83_2_1_1_2_3_2"/>
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p1">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c4">F</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</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">  ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Olive">b<sub>1</sub></font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">T</font>,<font color="Olive">b<sub>3</sub></font>)</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">K</font> =  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>);<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">E1</font> = <font color="Maroon" title="c5">E</font> as  <font color="Maroon" title="c4">F</font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>   <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <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="c4">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#T31" target="_self" onmouseover="rs('field_7/T31')" onmouseout="rh()">alg0</a>, <a class="ref" href="field_4.html#T7" onmouseover="rs('field_4/T7')" onmouseout="rh()">FIELD_4:7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">j</font> =  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c10">E1</font>,<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E4:83_2_1_1_2_3_2"/>
( <font color="Maroon" title="c11">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c11">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_2_1_1_2_3_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:83_2_1_1_2_3_2_1"/>
<font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:83_2_1_1_2_3_2_1"/>
<font color="Maroon" title="c11">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</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"> <font color="Maroon" title="c11">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font></span><br/>

<a NAME="E3:83_2_1_1_2_3_2_1"/><span class="lab"><font color="Green" title="E55">D7</font></span>: 
<font color="Maroon" title="c3">k</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="c10">E1</font>,<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>)</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> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">F</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_1"><span class="lab"><font color="Green" title="E50">Z</font></span></a>, <a class="ref" href="field_7.html#T30" target="_self" onmouseover="rs('field_7/T30')" onmouseout="rh()">degmult</a></span>;<br/>
<a NAME="E4:83_2_1_1_2_3_2_1"/><span class="lab"><font color="Green" title="E56">D6</font></span>: 
<font color="Maroon" title="c11">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_2_1_1_2_3_2_1"><span class="lab"><font color="Green" title="E55">D7</font></span></a>, <a class="ref" href="nat_1.html#T24" onmouseover="rs('nat_1/T24')" onmouseout="rh()">NAT_1:24</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_2_1_1_2_3_2_1_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">  not <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:83_2_1_1_2_3_2_1_1"/>
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</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"> contradiction</span><br/><span class="kw">then </span><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<span class="p3">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p4"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">F</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_1"><span class="lab"><font color="Green" title="E50">Z</font></span></a>, <a class="ref" href="field_7.html#T30" target="_self" onmouseover="rs('field_7/T30')" onmouseout="rh()">degmult</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">k</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> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">F</font>)</span>)</span>

;<br/><a NAME="E3:83_2_1_1_2_3_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <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> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">F</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_1_1_2"><span class="lab"><font color="Green" title="E51">B0</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>;<br/><span class="kw">hence </span><a NAME="E4:83_2_1_1_2_3_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_2_1_1_2"><span class="lab"><font color="Green" title="E54">C3</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</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="E6:83_2_1_1_2_3_2_1"/>
<font color="Maroon" title="c11">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_2_1_1_2_3_2_1"><span class="lab"><font color="Green" title="E56">D6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</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">then </span><span class="kw">consider </span><font color="Maroon" title="c12">T1</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c10">E1</font><span class="kw"> such that </span><br/><a NAME="E6:83_2_1_1_2_3_2"/><span class="lab"><font color="Green" title="E55">D3</font></span>: 
<font color="Maroon" title="c10">E1</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c12">T1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2"><span class="lab"><font color="Green" title="E49">AS</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">T</font> = <font color="Maroon" title="c12">T1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">E</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">a1</font> = <font color="Maroon" title="c7">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">E</font> ;<br/><a NAME="E9:83_2_1_1_2_3_2"/>
 for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">E1</font>  st <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">T</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> holds <br/><font color="Olive" title="b1">b</font> is <font color="Maroon" title="c4">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">T2</font> = <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c14">a1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c13">T</font> as  <font color="Maroon" title="c4">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">E</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#D12" target="_self" onmouseover="rs('field_7/D12')" onmouseout="rh()">defTalg</a></span>;<br/><a NAME="E11:83_2_1_1_2_3_2"/>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c12">T1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c15">T2</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#T35" target="_self" onmouseover="rs('field_7/T35')" onmouseout="rh()">ug1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:83_2_1_1_2_3_2"/>
 ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c4">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">E</font> st <font color="Maroon" title="c5">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c4">F</font>,<font color="Olive" title="b1">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_2_1_1_2_3_2"><span class="lab"><font color="Green" title="E55">D3</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:83_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</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><span class="kw">end;</span></div>
<a NAME="E3:83"/><span class="lab"><font color="Green" title="E49">I</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S9" onmouseover="rs('nat_1/S9')" onmouseout="rh()">NAT_1:sch 9</a>(<a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E48">IS</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E50">A</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_3"><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="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a>  implies  ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font> st <font color="Maroon" title="c2">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b1">T</font>) )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:83_3"/>
<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> 
 ; <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">  ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font> st <font color="Maroon" title="c2">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b1">T</font>)</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">k</font> =  <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>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E3:83_3"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:83_3"/>
 ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font> st <font color="Maroon" title="c2">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b1">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E49">I</font></span></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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_4"><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"> (  ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font> st <font color="Maroon" title="c2">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b1">T</font>) 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>  )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:83_4"/>
 ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font> st <font color="Maroon" title="c2">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b1">T</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">E</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">then </span><span class="kw">consider </span><font color="Maroon" title="c3">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E3:83_4"/><span class="lab"><font color="Green" title="E51">B0</font></span>: 
<font color="Maroon" title="c2">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c3">T</font>)
 ;<br/><a NAME="E4:83_4"/>
 <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c3">T</font>)</span>)</span>,<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="field_4.html#D8" onmouseover="rs('field_4/D8')" onmouseout="rh()">FIELD_4:def 8</a></span>;<br/><a NAME="E5:83_4"/><span class="kw">then </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="txt" href="#E3:83_4"><span class="lab"><font color="Green" title="E51">B0</font></span></a>, <a class="ref" href="field_7.html#T4" target="_self" onmouseover="rs('field_7/T4')" onmouseout="rh()">str12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:83_4"/>
<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="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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:83"/>
( <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>  iff  ex <font color="Olive" title="b1">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V3" title="FIELD_7:attr.3">-algebraic</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font> st <font color="Maroon" title="c2">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b1">T</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E50">A</font></span></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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