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<span class="kw">set </span><font color="Maroon" title="c3">E</font> =  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <span class="p2">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:117_1_1"/><span class="lab"><font color="Green" title="E52">A0</font></span>: 
<font color="Maroon" title="c1">F</font> is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <span class="p2">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 
<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="E2:117_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="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_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <span class="p4">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></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="c4">q</font> =  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</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_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <span class="p4">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span></span>)</span> ;<br/>
<span class="lab"><font color="Green" title="E53">A1</font></span>:  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c4">q</font>,<span class="p1">(<span class="default"><a href="field_5.html#K2" title="FIELD_5:func.2">KrRootP</a> <span class="p2">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>) = 
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p1">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T26" onmouseover="rs('field_4/T26')" onmouseout="rh()">FIELD_4:26</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#T17" onmouseover="rs('field_5/T17')" onmouseout="rh()">FIELD_5:17</a></span>
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="117_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">  not  <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <span class="p3">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:117_1_1_2"/>
 <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <span class="p3">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></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"> contradiction</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">o</font> =  <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_1_1"><span class="lab"><font color="Green" title="E52">A0</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>;<br/><a NAME="E3:117_1_1_2"/>
<font color="Maroon" title="c5">o</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a>  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117_1_1"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="field_4.html#T27" onmouseover="rs('field_4/T27')" onmouseout="rh()">FIELD_4:27</a></span>;<br/><a NAME="E4:117_1_1_2"/><span class="kw">then </span>
 <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font> is  <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> 
 ;<br/><span class="kw">hence </span><a NAME="E5:117_1_1_2"/>
contradiction
 ; <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="E6:117_1_1"/>
(  not  <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font> is <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6: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/>


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