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<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">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  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> holds <br/> <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>) is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a> ;<br/>
<a NAME="E1:81_1_1"/><span class="lab"><font color="Green" title="E46">X</font></span>: 
<font color="Maroon" title="c1">F</font> is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of <font color="Maroon" title="c2">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D1" onmouseover="rs('field_4/D1')" onmouseout="rh()">FIELD_4:def 1</a></span>;<br/>
<a NAME="E2:81_1_1"/><span class="lab"><font color="Green" title="E47">IA</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="81_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">T</font> be   <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>; <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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</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">assume </span><a NAME="E1:81_1_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>) is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a> </span><br/>

<a NAME="E2:81_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">AS</font></span>: 
<font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E3:81_1_1_1"/><span class="lab"><font color="Green" title="E49">Y</font></span>: 
 <a href="field_6.html#K4" title="FIELD_6:func.4">carrierFA</a> <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  for <font color="Olive" title="b2">U</font> being    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c2">E</font>  st <font color="Maroon" title="c1">F</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Olive" title="b2">U</font> &amp; <font color="Maroon" title="c4">T</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">U</font> holds <br/><font color="Olive" title="b1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">U</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#D5" onmouseover="rs('field_6/D5')" onmouseout="rh()">FIELD_6:def 5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">P</font> =  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">E1</font> = <font color="Maroon" title="c1">F</font> as   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T6" onmouseover="rs('field_4/T6')" onmouseout="rh()">FIELD_4:6</a></span>;<br/>
<a NAME="E5:81_1_1_1"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
 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="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">E1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="81_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E51">C1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="81_1_1_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">  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <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 <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="c4">T</font>)</span>)</span> 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="c6">E1</font></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 <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="c4">T</font>)</span>)</span> 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="c6">E1</font> )</span><br/><span class="kw">assume </span><a NAME="E1:81_1_1_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 <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="c4">T</font>)</span>)</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="c6">E1</font></span><br/><a NAME="E2:81_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="field_6.html#K4" title="FIELD_6:func.4">carrierFA</a> <font color="Maroon" title="c4">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#D6" onmouseover="rs('field_6/D6')" onmouseout="rh()">FIELD_6:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E4:81_1_1_1_1_1"/><span class="lab"><font color="Green" title="E52">C2</font></span>: 
( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp; (  for <font color="Olive" title="b1">U</font> being    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c2">E</font>  st <font color="Maroon" title="c1">F</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Olive" title="b1">U</font> &amp; <font color="Maroon" title="c4">T</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">U</font> holds <br/><font color="Maroon" title="c8">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">U</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1_1_1"><span class="lab"><font color="Green" title="E49">Y</font></span></a></span>;<br/><a NAME="E5:81_1_1_1_1_1"/>
( <font color="Maroon" title="c1">F</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c2">E</font> &amp; <font color="Maroon" title="c1">F</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c1">F</font> &amp; <font color="Maroon" title="c4">T</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">F</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>, <a class="ref" href="field_4.html#T1" onmouseover="rs('field_4/T1')" onmouseout="rh()">FIELD_4:1</a>, <a class="txt" href="#E2:81_1_1_1"><span class="lab"><font color="Green" title="E48">AS</font></span></a>, <a class="ref" href="xboole_1.html#T2" onmouseover="rs('xboole_1/T2')" onmouseout="rh()">XBOOLE_1:2</a></span>;<br/><a NAME="E6:81_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">C2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:81_1_1_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="c6">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="81_1_1_1_1_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">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <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="c1">F</font> 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 <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="c4">T</font>)</span>)</span></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="c1">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 <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="c4">T</font>)</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:81_1_1_1_1_2"/><span class="lab"><font color="Green" title="E52">C3</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="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="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 <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="c4">T</font>)</span>)</span></span><br/><a NAME="E2:81_1_1_1_1_2"/>
<font color="Maroon" title="c1">F</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T36" onmouseover="rs('field_6/T36')" onmouseout="rh()">FIELD_6:36</a></span>;<br/><a NAME="E3:81_1_1_1_1_2"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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 <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="c4">T</font>)</span>)</span>
 <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="E4:81_1_1_1_1_2"/>
<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 <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="c4">T</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">C3</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="E3:81_1_1_1_1"/>
 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="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_1_1"><span class="lab"><font color="Green" title="E51">C1</font></span></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="lab"><font color="Green" title="E51">B2</font></span>:  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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="c4">T</font>)</span>)</span> = 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#D6" onmouseover="rs('field_6/D6')" onmouseout="rh()">FIELD_6:def 6</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1"><span class="lab"><font color="Green" title="E46">X</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>
;<br/>
<span class="lab"><font color="Green" title="E52">B4</font></span>:  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span> = 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">E</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1">(<span class="default"><a href="field_6.html#K4" title="FIELD_6:func.4">carrierFA</a> <font color="Maroon" title="c4">T</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#D6" onmouseover="rs('field_6/D6')" onmouseout="rh()">FIELD_6:def 6</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">E</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81_1_1_1"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="field_6.html#D6" onmouseover="rs('field_6/D6')" onmouseout="rh()">FIELD_6:def 6</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1"><span class="lab"><font color="Green" title="E46">X</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>
;<br/>
<span class="lab"><font color="Green" title="E53">B5</font></span>:  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</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="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span> = 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">E</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1">(<span class="default"><a href="field_6.html#K4" title="FIELD_6:func.4">carrierFA</a> <font color="Maroon" title="c4">T</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#D6" onmouseover="rs('field_6/D6')" onmouseout="rh()">FIELD_6:def 6</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">E</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81_1_1_1"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="field_6.html#D6" onmouseover="rs('field_6/D6')" onmouseout="rh()">FIELD_6:def 6</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1"><span class="lab"><font color="Green" title="E46">X</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">h</font> =  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c6">E1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">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="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81_1_1_1"><span class="lab"><font color="Green" title="E50">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">P</font> =  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>) as  <font color="Maroon" title="c6">E1</font> <a href="ring_2.html#V3" title="RING_2:attr.3">-homomorphic</a>  <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font> ;<br/>
<a NAME="E11:81_1_1_1"/>
( <font color="Maroon" title="c7">h</font> is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a>  &amp; <font color="Maroon" title="c7">h</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Maroon" title="c7">h</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:81_1_1_1"><span class="lab"><font color="Green" title="E52">B4</font></span></a>, <a class="txt" href="#E8:81_1_1_1"><span class="lab"><font color="Green" title="E53">B5</font></span></a>, <a class="txt" href="#E6:81_1_1_1"><span class="lab"><font color="Green" title="E51">B2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c7">h</font> as   <a href="ring_2.html#NM2" title="RING_2:NM.2">Homomorphism</a> of <font color="Maroon" title="c6">E1</font>,<font color="Maroon" title="c8">P</font> ;<br/>
<a NAME="E13:81_1_1_1"/>
 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">E1</font>  st <font color="Olive" title="b1">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Maroon" title="c9">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c9">h</font> as   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c6">E1</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E48"><span class="lab"><font color="Green" title="E39">lintrans</font></span></a></span>;<br/>
<a NAME="E15:81_1_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>
 
;<br/>
<a NAME="E16:81_1_1_1"/><span class="kw">then </span>
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c6">E1</font>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#T7" target="_self" onmouseover="rs('field_7/T7')" onmouseout="rh()">quah1</a></span>;<br/>
<a NAME="E17:81_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c6">E1</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> is   <a href="vectsp_7.html#NM1" title="VECTSP_7:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c6">E1</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#T9" target="_self" onmouseover="rs('field_7/T9')" onmouseout="rh()">quah2</a></span>;<br/>
<a NAME="E18:81_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A2</font></span>: 
 <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c6">E1</font>,<font color="Maroon" title="c1">F</font>) is  <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D1" onmouseover="rs('matrlin/D1')" onmouseout="rh()">MATRLIN:def 1</a></span>;<br/>
<a NAME="E19:81_1_1_1"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c6">E1</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">E1</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</a></span>;<br/>
<a NAME="E20:81_1_1_1"/>
<font color="Maroon" title="c10">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81_1_1_1"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E19:81_1_1_1"><span class="lab"><font color="Green" title="E55">A3</font></span></a></span>;<br/>
<a NAME="E21:81_1_1_1"/><span class="kw">then </span>
 <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c8">P</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="#E18:81_1_1_1"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="vectsp12.html#T4" onmouseover="rs('vectsp12/T4')" onmouseout="rh()">VECTSP12:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:81_1_1_1"/>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</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>
<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="81_1_1_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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:81_1_1_2"/><span class="lab"><font color="Green" title="E49">IV</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="81_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">  for <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  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> <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>) 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">let </span><font color="Maroon" title="c5">T</font> be   <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>; <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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c5">T</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">assume </span><a NAME="E1:81_1_1_2_1"/><span class="lab"><font color="Green" title="E50">AS</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.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">  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c5">T</font>) is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a> </span><br/><span class="kw">set </span><font color="Maroon" title="c6">a</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">T</font>;<br/><a NAME="E2:81_1_1_2_1"/><span class="lab"><font color="Green" title="E51">I</font></span>: 
<font color="Maroon" title="c5">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_2_1"><span class="lab"><font color="Green" title="E50">AS</font></span></a></span>;<br/><a NAME="E3:81_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">H</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">T</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">T</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c8">T1</font> = <font color="Maroon" title="c5">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="81_1_1_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">  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="c2">E</font>  st <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</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="c9">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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> implies <font color="Maroon" title="c9">b</font> is <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:81_1_1_2_1_1"/>
<font color="Maroon" title="c9">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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>
 ; <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="c9">b</font> is <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> </span><br/><a NAME="E2:81_1_1_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">T</font> &amp;  not <font color="Maroon" title="c9">b</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="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:81_1_1_2_1_1"/>
<font color="Maroon" title="c9">b</font> is <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> 
 <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>; <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="E6:81_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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> is <font color="Maroon" title="c1">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="c9">T1</font> = <font color="Maroon" title="c5">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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> as   <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> ;<br/><a NAME="E8:81_1_1_2_1"/><span class="lab"><font color="Green" title="E53">A0</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" onmouseover="rs('xboole_1/T39')" onmouseout="rh()">XBOOLE_1:39</a></span>;<br/><a NAME="E9:81_1_1_2_1"/>
<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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1_1_2_1"><span class="lab"><font color="Green" title="E52">H</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E10:81_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A1</font></span>: 
<font color="Maroon" title="c5">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:81_1_1_2_1"><span class="lab"><font color="Green" title="E53">A0</font></span></a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="81_1_1_2_1_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">  not <font color="Maroon" title="c7">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">T1</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:81_1_1_2_1_2"/>
<font color="Maroon" title="c7">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">T1</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="E2:81_1_1_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">T</font> &amp;  not <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="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:81_1_1_2_1_2"/>
contradiction
 <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>; <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="E12:81_1_1_2_1"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">T1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:81_1_1_2_1"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="card_2.html#T41" onmouseover="rs('card_2/T41')" onmouseout="rh()">CARD_2:41</a></span>;<br/><a NAME="E13:81_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A3</font></span>: 
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c9">T1</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="#E1:81_1_1_2_1"><span class="lab"><font color="Green" title="E50">AS</font></span></a>, <a class="txt" href="#E1:81_1_1_2"><span class="lab"><font color="Green" title="E49">IV</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">E1</font> = <font color="Maroon" title="c2">E</font> as   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c9">T1</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/><span class="kw">reconsider </span><font color="Maroon" title="c11">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="c10">E1</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c12">p</font> being   non  <a href="ratfunc1.html#NV1" title="RATFUNC1:NV.1">zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E17:81_1_1_2_1"/><span class="lab"><font color="Green" title="E56">H1</font></span>: 
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c7">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1_1_2_1"><span class="lab"><font color="Green" title="E51">I</font></span></a>, <a class="ref" href="field_7.html#D12" target="_self" onmouseover="rs('field_7/D12')" onmouseout="rh()">defTalg</a>, <a class="ref" href="field_6.html#T43" onmouseover="rs('field_6/T43')" onmouseout="rh()">FIELD_6:43</a></span>;<br/><a NAME="E18:81_1_1_2_1"/><span class="lab"><font color="Green" title="E57">H3</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">F</font>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">p</font> = <font color="Maroon" title="c12">p</font> as   non  <a href="ratfunc1.html#NV1" title="RATFUNC1:NV.1">zero</a>   <a href="ring_4.html#M1" title="RING_4:mode.1">Element</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="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D10" onmouseover="rs('polynom3/D10')" onmouseout="rh()">POLYNOM3:def 10</a></span>;<br/><a NAME="E20:81_1_1_2_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<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="c9">T1</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T10" onmouseover="rs('field_4/T10')" onmouseout="rh()">FIELD_4:10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">p1</font> = <font color="Maroon" title="c13">p</font> as    <a href="ring_4.html#M1" title="RING_4:mode.1">Element</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="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<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="c9">T1</font>)</span>)</span></span>)</span> ;<br/><a NAME="E22:81_1_1_2_1"/><span class="lab"><font color="Green" title="E58">H4</font></span>: 
<font color="Maroon" title="c13">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</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="c9">T1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:81_1_1_2_1"><span class="lab"><font color="Green" title="E57">H3</font></span></a>, <a class="ref" href="field_4.html#T12" onmouseover="rs('field_4/T12')" onmouseout="rh()">FIELD_4:12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">p1</font> = <font color="Maroon" title="c14">p1</font> as   non  <a href="ratfunc1.html#NV1" title="RATFUNC1:NV.1">zero</a>   <a href="ring_4.html#M1" title="RING_4:mode.1">Element</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="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<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="c9">T1</font>)</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:81_1_1_2_1"><span class="lab"><font color="Green" title="E58">H4</font></span></a>, <a class="ref" href="uproots.html#D5" onmouseover="rs('uproots/D5')" onmouseout="rh()">UPROOTS:def 5</a></span>;<br/><a NAME="E24:81_1_1_2_1"/>
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c15">p1</font>,<font color="Maroon" title="c11">a1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:81_1_1_2_1"><span class="lab"><font color="Green" title="E56">H1</font></span></a>, <a class="ref" href="field_7.html#T15" target="_self" onmouseover="rs('field_7/T15')" onmouseout="rh()">lemma7b</a></span>;<br/><a NAME="E25:81_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<font color="Maroon" title="c11">a1</font> is  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c9">T1</font>) <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T43" onmouseover="rs('field_6/T43')" onmouseout="rh()">FIELD_6:43</a></span>;<br/><a NAME="E26:81_1_1_2_1"/>
 <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="c1">F</font>,<font color="Maroon" title="c9">T1</font>)</span>)</span>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">a1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <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="c1">F</font>,<font color="Maroon" title="c5">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:81_1_1_2_1"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="field_7.html#T34" target="_self" onmouseover="rs('field_7/T34')" onmouseout="rh()">ug</a></span>;<br/><span class="kw">hence </span><a NAME="E27:81_1_1_2_1"/>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c5">T</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="#E13:81_1_1_2_1"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E25:81_1_1_2_1"><span class="lab"><font color="Green" title="E59">A4</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="E3:81_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.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"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:81_1_1"/><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> holds  <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#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E2:81_1_1"><span class="lab"><font color="Green" title="E47">IA</font></span></a>, <a class="txt" href="#E3:81_1_1"><span class="lab"><font color="Green" title="E48">IS</font></span></a>);<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:81_1_1"/><span class="lab"><font color="Green" title="E50">H</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 ;<br/>
<span class="kw">thus </span><a NAME="E7:81_1_1"/>
 <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>) 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="#E4:81_1_1"><span class="lab"><font color="Green" title="E49">I</font></span></a>, <a class="txt" href="#E6:81_1_1"><span class="lab"><font color="Green" title="E50">H</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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