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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   non 2 <a href="ring_3.html#V3" title="RING_3:attr.3">-characteristic</a>   <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a>   non  <a href="field_9.html#V3" title="FIELD_9:attr.3">quadratic_complete</a>  <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">p</font> being   <a href="field_9.html#V1" title="FIELD_9:attr.1">quadratic</a>   non  <a href="field_9.html#V4" title="FIELD_9:attr.4">DC-square</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> holds   <a href="field_4.html#K1" title="FIELD_4:func.1">Roots</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p4">(<span class="default"><a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Olive" title="b1">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_7.html#K1" title="FIELD_7:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K18" title="FIELD_9:func.18">Root1</a> <font color="Olive" title="b1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="field_9.html#K19" title="FIELD_9:func.19">Root2</a> <font color="Olive" title="b1">p</font></span>)</span></span><a href="field_7.html#K1" title="FIELD_7:func.1">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="field_9.html#V1" title="FIELD_9:attr.1">quadratic</a>   non  <a href="field_9.html#V4" title="FIELD_9:attr.4">DC-square</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>; <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#K1" title="FIELD_4:func.1">Roots</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p4">(<span class="default"><a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c2">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_7.html#K1" title="FIELD_7:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K18" title="FIELD_9:func.18">Root1</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="field_9.html#K19" title="FIELD_9:func.19">Root2</a> <font color="Maroon" title="c2">p</font></span>)</span></span><a href="field_7.html#K1" title="FIELD_7:func.1">}</a></span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">E</font> =  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<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="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p3">(<span class="default"><a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">r1</font> =  <a href="field_9.html#K18" title="FIELD_9:func.18">Root1</a> <font color="Maroon" title="c2">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">r2</font> =  <a href="field_9.html#K19" title="FIELD_9:func.19">Root2</a> <font color="Maroon" title="c2">p</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">q</font> = <span class="p1">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<span class="p2">(<span class="default"><a href="ratfunc1.html#NK3" title="RATFUNC1:NK.3">LC</a> <font color="Maroon" title="c2">p</font></span>)</span>,<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>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p5">(<span class="default"><a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K2" title="FIELD_9:func.2">X-</a> <span class="p3">(<span class="default"><a href="field_9.html#K18" title="FIELD_9:func.18">Root1</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2">(<span class="default"><a href="field_9.html#K2" title="FIELD_9:func.2">X-</a> <span class="p3">(<span class="default"><a href="field_9.html#K19" title="FIELD_9:func.19">Root2</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> 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>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p5">(<span class="default"><a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></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="E2:144"/><span class="lab"><font color="Green" title="E66">K</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_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<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="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p3">(<span class="default"><a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></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="field_4.html#D1" onmouseover="rs('field_4/D1')" onmouseout="rh()">FIELD_4:def 1</a></span>;<br/>
<a NAME="E3:144"/><span class="lab"><font color="Green" title="E67">Y</font></span>: 
 <a href="field_4.html#K1" title="FIELD_4:func.1">Roots</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p4">(<span class="default"><a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c2">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c6">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T65" target="_self" onmouseover="rs('field_9/T65')" onmouseout="rh()">Z5</a>, <a class="ref" href="field_7.html#T13" onmouseover="rs('field_7/T13')" onmouseout="rh()">FIELD_7:13</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E68">Z</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="144_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  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<span class="p1">(<span class="default"><a href="ratfunc1.html#NK3" title="RATFUNC1:NK.3">LC</a> <font color="Maroon" title="c2">p</font></span>)</span>,<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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p4">(<span class="default"><a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>) is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:144_1"/><span class="lab"><font color="Green" title="E69">Y1</font></span>: 
 <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<span class="p1">(<span class="default"><a href="ratfunc1.html#NK3" title="RATFUNC1:NK.3">LC</a> <font color="Maroon" title="c2">p</font></span>)</span>,<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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p4">(<span class="default"><a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>) is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</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"> contradiction</span><br/> <a href="ratfunc1.html#NK3" title="RATFUNC1:NK.3">LC</a> <font color="Maroon" title="c2">p</font> = 
 <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<span class="p1">(<span class="default"><a href="ratfunc1.html#NK3" title="RATFUNC1:NK.3">LC</a> <font color="Maroon" title="c2">p</font></span>)</span>,<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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p4">(<span class="default"><a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></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_7.html#D4" onmouseover="rs('field_7/D4')" onmouseout="rh()">FIELD_7:def 4</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="txt" href="#E1:144_1"><span class="lab"><font color="Green" title="E69">Y1</font></span></a>, <a class="txt" href="#E2:144"><span class="lab"><font color="Green" title="E66">K</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="E3:144_1"/>
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>
<a NAME="E5:144"/>
(  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p2">(<span class="default"><a href="field_9.html#K18" title="FIELD_9:func.18">Root1</a> <font color="Maroon" title="c2">p</font></span>)</span>)</span>)</span> <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"><span class="p2">(<span class="default"><a href="field_9.html#K18" title="FIELD_9:func.18">Root1</a> <font color="Maroon" title="c2">p</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> &amp;  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p2">(<span class="default"><a href="field_9.html#K19" title="FIELD_9:func.19">Root2</a> <font color="Maroon" title="c2">p</font></span>)</span>)</span>)</span> <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"><span class="p2">(<span class="default"><a href="field_9.html#K19" title="FIELD_9:func.19">Root2</a> <font color="Maroon" title="c2">p</font></span>)</span></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="ring_5.html#T18" onmouseover="rs('ring_5/T18')" onmouseout="rh()">RING_5:18</a></span>;<br/>
<span class="kw">then </span> <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p3">(<span class="default"><a href="field_9.html#K18" title="FIELD_9:func.18">Root1</a> <font color="Maroon" title="c2">p</font></span>)</span>)</span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p3">(<span class="default"><a href="field_9.html#K19" title="FIELD_9:func.19">Root2</a> <font color="Maroon" title="c2">p</font></span>)</span>)</span>)</span></span>)</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="field_9.html#K18" title="FIELD_9:func.18">Root1</a> <font color="Maroon" title="c2">p</font></span>)</span></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> <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="field_9.html#K19" title="FIELD_9:func.19">Root2</a> <font color="Maroon" title="c2">p</font></span>)</span></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="uproots.html#T23" onmouseover="rs('uproots/T23')" onmouseout="rh()">UPROOTS:23</a></span>
<br/>.= 
<span class="p1"><a href="field_7.html#K1" title="FIELD_7:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K18" title="FIELD_9:func.18">Root1</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="field_9.html#K19" title="FIELD_9:func.19">Root2</a> <font color="Maroon" title="c2">p</font></span>)</span></span><a href="field_7.html#K1" title="FIELD_7:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T1" onmouseover="rs('enumset1/T1')" onmouseout="rh()">ENUMSET1:1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:144"/>
 <a href="field_4.html#K1" title="FIELD_4:func.1">Roots</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p4">(<span class="default"><a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c2">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_7.html#K1" title="FIELD_7:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K18" title="FIELD_9:func.18">Root1</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="field_9.html#K19" title="FIELD_9:func.19">Root2</a> <font color="Maroon" title="c2">p</font></span>)</span></span><a href="field_7.html#K1" title="FIELD_7:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:144"><span class="lab"><font color="Green" title="E67">Y</font></span></a>, <a class="txt" href="#E4:144"><span class="lab"><font color="Green" title="E68">Z</font></span></a>, <a class="ref" href="ring_5.html#T19" onmouseover="rs('ring_5/T19')" onmouseout="rh()">RING_5:19</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>
