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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><br/>  for <font color="Olive" title="b2">K</font> being  <font color="Olive">b<sub>1</sub></font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>  <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>  st <font color="Olive" title="b2">K</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a>  holds <br/>( <font color="Olive" title="b2">K</font> is <font color="Olive" title="b1">E</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a>  &amp; <font color="Olive" title="b1">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</font> be   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being  <font color="Maroon" title="c2">E</font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>  <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>  st <font color="Olive" title="b1">K</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a>  holds <br/>( <font color="Olive" title="b1">K</font> is <font color="Maroon" title="c2">E</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a>  &amp; <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font> be  <font color="Maroon" title="c2">E</font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>  <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">K</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a>  implies ( <font color="Maroon" title="c3">K</font> is <font color="Maroon" title="c2">E</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a>  &amp; <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a>  ) )</span><br/>





<a NAME="E1:86"/><span class="lab"><font color="Green" title="E47">H0</font></span>: 
( <font color="Maroon" title="c2">E</font> is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of <font color="Maroon" title="c3">K</font> &amp; <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="E2:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">H1</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</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="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:86"/><span class="lab"><font color="Green" title="E49">AS</font></span>: 
<font color="Maroon" title="c3">K</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</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> is <font color="Maroon" title="c2">E</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a>  &amp; <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a>  )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="86_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">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font> holds  <font color="Olive" title="b1">a</font> is <font color="Maroon" title="c2">E</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="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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"> <font color="Maroon" title="c4">a</font> is <font color="Maroon" title="c2">E</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c5">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="E2:86_1"/><span class="lab"><font color="Green" title="E50">A</font></span>: 
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">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="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86"><span class="lab"><font color="Green" title="E49">AS</font></span></a>, <a class="ref" href="field_6.html#T43" onmouseover="rs('field_6/T43')" onmouseout="rh()">FIELD_6:43</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">p1</font> = <font color="Maroon" title="c5">p</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c2">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E47">H0</font></span></a>, <a class="ref" href="field_4.html#T9" onmouseover="rs('field_4/T9')" onmouseout="rh()">FIELD_4:9</a></span>;<br/><a NAME="E4:86_1"/>
(  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c2">E</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">F</font> &amp; <font color="Maroon" title="c5">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T12" onmouseover="rs('field_4/T12')" onmouseout="rh()">FIELD_4:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">p1</font> = <font color="Maroon" title="c6">p1</font> as   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="c2">E</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D5" onmouseover="rs('uproots/D5')" onmouseout="rh()">UPROOTS:def 5</a></span>;<br/><a NAME="E6:86_1"/>
( <font color="Maroon" title="c5">p</font> is    <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> &amp; <font color="Maroon" title="c7">p1</font> is    <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="c2">E</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/><span class="kw">then </span> <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c4">a</font>) = 
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">a</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#T15" target="_self" onmouseover="rs('field_7/T15')" onmouseout="rh()">lemma7b</a></span>
<br/>.= 
 <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="#E2:86_1"><span class="lab"><font color="Green" title="E50">A</font></span></a>, <a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E47">H0</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">hence </span><a NAME="E8:86_1"/>
<font color="Maroon" title="c4">a</font> is <font color="Maroon" title="c2">E</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E48">H1</font></span></a>, <a class="ref" href="field_6.html#T43" onmouseover="rs('field_6/T43')" onmouseout="rh()">FIELD_6:43</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="E5:86"/>
<font color="Maroon" title="c3">K</font> is <font color="Maroon" title="c2">E</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="86_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">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> holds  <font color="Olive" title="b1">a</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="c4">a</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="c4">a</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">consider </span><font color="Maroon" title="c5">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="E2:86_2"/><span class="lab"><font color="Green" title="E50">A</font></span>: 
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">K</font>)</span>)</span>) <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="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86"><span class="lab"><font color="Green" title="E49">AS</font></span></a>, <a class="ref" href="field_6.html#T43" onmouseover="rs('field_6/T43')" onmouseout="rh()">FIELD_6:43</a></span>;<br/><a NAME="E3:86_2"/>
<font color="Maroon" title="c5">p</font> is    <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/><span class="kw">then </span> <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">a</font>) = 
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">K</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T11" onmouseover="rs('field_6/T11')" onmouseout="rh()">FIELD_6:11</a></span>
<br/>.= 
 <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="#E2:86_2"><span class="lab"><font color="Green" title="E50">A</font></span></a>, <a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E47">H0</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">hence </span><a NAME="E5:86_2"/>
<font color="Maroon" title="c4">a</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_6.html#T43" onmouseover="rs('field_6/T43')" onmouseout="rh()">FIELD_6:43</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="E7:86"/>
<font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</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"> verum</span><br/>


</div>
