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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_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="Olive" title="b1">p</font></span>)</span></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of <font color="Olive" title="b1">p</font></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_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>) is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of <font color="Maroon" title="c2">p</font></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="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="polynom3.html#K11" title="POLYNOM3:func.11">*'</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> 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="E2:146"/><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/>
<div><span class="lab"><font color="Green" title="E67">Y</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_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="E1:146_1"/><span class="lab"><font color="Green" title="E68">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="#E1:146_1"><span class="lab"><font color="Green" title="E68">Y1</font></span></a>, <a class="txt" href="#E2:146"><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="E3:146_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="E4:146"/>
<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="polynom3.html#K11" title="POLYNOM3:func.11">*'</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="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="field_9.html#K2" title="FIELD_9:func.2">X-</a> <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="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p1">(<span class="default"><a href="field_9.html#K2" title="FIELD_9:func.2">X-</a> <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>
 
;<br/>
<a NAME="E5:146"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">D</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <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="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="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></span>;<br/>
<a NAME="E6:146"/>
(  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p1">(<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>) is    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</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>) &amp;  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p1">(<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>) is    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</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="ring_5.html#T51" onmouseover="rs('ring_5/T51')" onmouseout="rh()">RING_5:51</a></span>;<br/>
<a NAME="E7:146"/><span class="kw">then </span>
<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="polynom3.html#K11" title="POLYNOM3:func.11">*'</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> is    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</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="ring_5.html#T52" onmouseover="rs('ring_5/T52')" onmouseout="rh()">RING_5:52</a></span>;<br/>
<a NAME="E8:146"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a>  <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="txt" href="#E3:146"><span class="lab"><font color="Green" title="E67">Y</font></span></a>, <a class="txt" href="#E5:146"><span class="lab"><font color="Green" title="E68">D</font></span></a>, <a class="ref" href="field_4.html#D5" onmouseover="rs('field_4/D5')" onmouseout="rh()">FIELD_4:def 5</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_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">U</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>  st <font color="Maroon" title="c2">p</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b1">U</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>,<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>) holds <br/><font color="Olive" title="b1">U</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<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></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">U</font> be   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Maroon" title="c7">U</font> &amp; <font color="Maroon" title="c7">U</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>,<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>) implies <font color="Maroon" title="c7">U</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<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><br/><span class="kw">assume </span><a NAME="E1:146_2"/><span class="lab"><font color="Green" title="E70">D0</font></span>: 
( <font color="Maroon" title="c2">p</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Maroon" title="c7">U</font> &amp; <font color="Maroon" title="c7">U</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>,<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>) )
 ; <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">U</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<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><br/><a NAME="E2:146_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">D3</font></span>: 
(  <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>) is <font color="Maroon" title="c7">U</font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>  &amp; <font color="Maroon" title="c7">U</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_5.html#T12" onmouseover="rs('field_5/T12')" onmouseout="rh()">FIELD_5:12</a>, <a class="ref" href="field_4.html#T7" onmouseover="rs('field_4/T7')" onmouseout="rh()">FIELD_4:7</a></span>;<br/><a NAME="E3:146_2"/><span class="lab"><font color="Green" title="E72">D4</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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:146_2"><span class="lab"><font color="Green" title="E71">D3</font></span></a>, <a class="txt" href="#E1:146_2"><span class="lab"><font color="Green" title="E70">D0</font></span></a>, <a class="txt" href="#E8:146"><span class="lab"><font color="Green" title="E69">A</font></span></a>, <a class="ref" href="field_8.html#T27" onmouseover="rs('field_8/T27')" onmouseout="rh()">FIELD_8:27</a></span>;<br/><a NAME="E4:146_2"/>
 <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="ref" href="field_9.html#T66" target="_self" onmouseover="rs('field_9/T66')" onmouseout="rh()">Z4</a></span>;<br/><a NAME="E5:146_2"/><span class="kw">then </span>
 <a href="field_9.html#K18" title="FIELD_9:func.18">Root1</a> <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">w</font> =  <a href="field_9.html#K18" title="FIELD_9:func.18">Root1</a> <font color="Maroon" title="c2">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">U</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:146_2"><span class="lab"><font color="Green" title="E72">D4</font></span></a></span>;<br/><a NAME="E7:146_2"/>
 <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p1">(<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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c9">aF</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c10">bF</font>, <font color="Maroon" title="c11">cF</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E2:146_2_1"/><span class="lab"><font color="Green" title="E73">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c11">cF</font>,<font color="Maroon" title="c10">bF</font>,<font color="Maroon" title="c9">aF</font></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T21" target="_self" onmouseover="rs('field_9/T21')" onmouseout="rh()">qua5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">aE</font> =  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</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>,<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>);<br/>
<span class="kw">set </span><font color="Maroon" title="c13">bE</font> =  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c10">bF</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>,<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>);<br/>
<span class="kw">set </span><font color="Maroon" title="c14">cE</font> =  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c11">cF</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>,<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>);<br/>
<div><span class="lab"><font color="Green" title="E74">C1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_2_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</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>,<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="E1:146_2_1_1"/><span class="lab"><font color="Green" title="E75">C2</font></span>: 
 <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</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>,<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/><font color="Maroon" title="c9">aF</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">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>,<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="txt" href="#E1:146_2_1_1"><span class="lab"><font color="Green" title="E75">C2</font></span></a>, <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="#E2:146"><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="E3:146_2_1_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="E4:146_2_1"/>
 not  <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>) is 2 <a href="ring_3.html#V3" title="RING_3:attr.3">-characteristic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:146"><span class="lab"><font color="Green" title="E66">K</font></span></a></span>;<br/>
<a NAME="E5:146_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">C3</font></span>: 
2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<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="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">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>,<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="txt" href="#E3:146_2_1"><span class="lab"><font color="Green" title="E74">C1</font></span></a>, <a class="ref" href="field_9.html#T5" target="_self" onmouseover="rs('field_9/T5')" onmouseout="rh()">ch2</a></span>;<br/>
<a NAME="E6:146_2_1"/><span class="lab"><font color="Green" title="E76">I1</font></span>: 
(  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</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>,<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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">aF</font> &amp;  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c10">bF</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>,<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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">bF</font> &amp;  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c11">cF</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>,<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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">cF</font> )
 
<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 NAME="E7:146_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c11">cF</font>,<span class="p3">(<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="p4"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p0">(<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="p2">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c10">bF</font>,<span class="p3">(<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="p4"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p0">(<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="p2">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<span class="p3">(<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="p4"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p0">(<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><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:146_2_1"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="ref" href="field_9.html#T23" target="_self" onmouseover="rs('field_9/T23')" onmouseout="rh()">eval2</a></span>;<br/>
<a NAME="E8:146_2_1"/><span class="kw">then </span>
 <a href="field_9.html#K18" title="FIELD_9:func.18">Root1</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c10">bF</font>,<span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p0">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p1">(<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>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="field_9.html#K17" title="FIELD_9:func.17">RootDC</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p0">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p1">(<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>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:146_2_1"><span class="lab"><font color="Green" title="E74">C1</font></span></a>, <a class="ref" href="field_9.html#T64" target="_self" onmouseover="rs('field_9/T64')" onmouseout="rh()">Z2</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<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> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<span class="p3">(<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="p4"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p0">(<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>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c10">bF</font>,<span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p0">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p1">(<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>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="field_9.html#K17" title="FIELD_9:func.17">RootDC</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<span class="p5">(<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="p0"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p1">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p2">(<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>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p0">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p1">(<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>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c10">bF</font>,<span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p0">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p1">(<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>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="field_9.html#K17" title="FIELD_9:func.17">RootDC</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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="txt" href="#E5:146_2_1"><span class="lab"><font color="Green" title="E75">C3</font></span></a>, <a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E77">B</font></span>: <span class="p1">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c10">bF</font>,<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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p0">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p1">(<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>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c10">bF</font>,<span class="p3">(<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="p4"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p0">(<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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c10">bF</font>,<span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p0">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p1">(<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>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="field_9.html#K17" title="FIELD_9:func.17">RootDC</a> <font color="Maroon" title="c2">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</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> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="field_9.html#K17" title="FIELD_9:func.17">RootDC</a> <font color="Maroon" title="c2">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T5" onmouseover="rs('rlvect_1/T5')" onmouseout="rh()">RLVECT_1:5</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">aU</font> =  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<font color="Maroon" title="c7">U</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c16">bU</font> =  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c10">bF</font>,<font color="Maroon" title="c7">U</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c17">cU</font> =  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c11">cF</font>,<font color="Maroon" title="c7">U</font>);<br/>
<a NAME="E11:146_2_1"/><span class="lab"><font color="Green" title="E78">I2</font></span>: 
(  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<font color="Maroon" title="c7">U</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">aF</font> &amp;  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c10">bF</font>,<font color="Maroon" title="c7">U</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">bF</font> &amp;  <a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c11">cF</font>,<font color="Maroon" title="c7">U</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">cF</font> )
 
<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 NAME="E12:146_2_1"/>
2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<font color="Maroon" title="c7">U</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<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="txt" href="#E2:146_2"><span class="lab"><font color="Green" title="E71">D3</font></span></a>, <a class="txt" href="#E11:146_2_1"><span class="lab"><font color="Green" title="E78">I2</font></span></a>, <a class="ref" href="field_9.html#T7" target="_self" onmouseover="rs('field_9/T7')" onmouseout="rh()">Z3</a>, <a class="ref" href="field_7.html#D4" onmouseover="rs('field_7/D4')" onmouseout="rh()">FIELD_7:def 4</a></span>;<br/>
<a NAME="E13:146_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<font color="Maroon" title="c7">U</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<span class="p3">(<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="p4"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p0">(<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>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:146_2"><span class="lab"><font color="Green" title="E71">D3</font></span></a>, <a class="ref" href="field_6.html#T16" onmouseover="rs('field_6/T16')" onmouseout="rh()">FIELD_6:16</a></span>;<br/>
<a NAME="E14:146_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c10">bF</font>,<font color="Maroon" title="c7">U</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<font color="Maroon" title="c7">U</font>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c10">bF</font>,<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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c9">aF</font>,<span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p0">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p1">(<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>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:146_2"><span class="lab"><font color="Green" title="E71">D3</font></span></a>, <a class="txt" href="#E6:146_2_1"><span class="lab"><font color="Green" title="E76">I1</font></span></a>, <a class="txt" href="#E11:146_2_1"><span class="lab"><font color="Green" title="E78">I2</font></span></a>, <a class="ref" href="field_6.html#T15" onmouseover="rs('field_6/T15')" onmouseout="rh()">FIELD_6:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:146_2_1"/>
 <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <span class="p1">(<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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:146_2_1"><span class="lab"><font color="Green" title="E77">B</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:146_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">D1</font></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#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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E9:146_2"/><span class="lab"><font color="Green" title="E74">D2</font></span>: 
<font color="Maroon" title="c7">U</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</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> <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>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:146_2"><span class="lab"><font color="Green" title="E70">D0</font></span></a>, <a class="ref" href="ec_pf_1.html#T5" onmouseover="rs('ec_pf_1/T5')" onmouseout="rh()">EC_PF_1:5</a></span>;<br/><a NAME="E10:146_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 <font color="Maroon" title="c7">U</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T7" onmouseover="rs('field_4/T7')" onmouseout="rh()">FIELD_4:7</a></span>;<br/><a NAME="E11:146_2"/><span class="kw">then </span>
 <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>) is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c7">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:146_2"><span class="lab"><font color="Green" title="E73">D1</font></span></a>, <a class="txt" href="#E9:146_2"><span class="lab"><font color="Green" title="E74">D2</font></span></a>, <a class="ref" href="field_6.html#T37" onmouseover="rs('field_6/T37')" onmouseout="rh()">FIELD_6:37</a></span>;<br/><span class="kw">hence </span><a NAME="E12:146_2"/>
<font color="Maroon" title="c7">U</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<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="txt" href="#E1:146_2"><span class="lab"><font color="Green" title="E70">D0</font></span></a>, <a class="ref" href="field_7.html#D2" onmouseover="rs('field_7/D2')" onmouseout="rh()">FIELD_7:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:146"/>
 <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>) is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:146"><span class="lab"><font color="Green" title="E69">A</font></span></a>, <a class="ref" href="field_8.html#D1" onmouseover="rs('field_8/D1')" onmouseout="rh()">FIELD_8: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/>


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