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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="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>   <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 <br/> ( <font color="Maroon" title="c1">F</font> is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of <font color="Olive" title="b1">p</font> iff  <a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Olive" title="b1">p</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  )</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>   <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"> ( <font color="Maroon" title="c1">F</font> is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of <font color="Maroon" title="c2">p</font> iff  <a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">a</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="c4">b</font>, <font color="Maroon" title="c5">c</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="E2:137"/><span class="lab"><font color="Green" title="E62">A</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="c5">c</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</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/>
<div><span class="lab"><font color="Green" title="E63">B</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="137_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"> (  <a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  implies <font color="Maroon" title="c1">F</font> is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of <font color="Maroon" title="c2">p</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:137_1"/>
 <a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">F</font> is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of <font color="Maroon" title="c2">p</font></span><br/><a NAME="E2:137_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c</font></span>)</span> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:137"><span class="lab"><font color="Green" title="E62">A</font></span></a>, <a class="ref" href="field_9.html#D9" target="_self" onmouseover="rs('field_9/D9')" onmouseout="rh()">defDC</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">w</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="E4:137_1"/><span class="lab"><font color="Green" title="E64">C</font></span>: 
<font color="Maroon" title="c6">w</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="o_ring_1.html#D2" onmouseover="rs('o_ring_1/D2')" onmouseout="rh()">O_RING_1:def 2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">r1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">w</font></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> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c8">r2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">w</font></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> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>;<br/><a NAME="E5:137_1"/><span class="lab"><font color="Green" title="E65">D</font></span>: 
<span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">a</font> <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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:137_1"><span class="lab"><font color="Green" title="E64">C</font></span></a>, <a class="ref" href="field_9.html#T28" target="_self" onmouseover="rs('field_9/T28')" onmouseout="rh()">lemred</a></span>;<br/><a NAME="E6:137_1"/>
(  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <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> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span>) is    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of <font color="Maroon" title="c1">F</font> &amp;  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <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> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span>) is    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of <font color="Maroon" title="c1">F</font> )
 <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="E7:137_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">E</font></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span>)</span>)</span> is    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of <font color="Maroon" title="c1">F</font>
 <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="E8:137_1"/><span class="lab"><font color="Green" title="E67">F</font></span>: 
<font color="Maroon" title="c1">F</font> is   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T6" onmouseover="rs('field_4/T6')" onmouseout="rh()">FIELD_4:6</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="137_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</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">E</font> &amp; <font color="Olive" title="b1">E</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c1">F</font> holds <br/><font color="Olive" title="b1">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a> <font color="Maroon" title="c1">F</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">E</font> be   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c9">E</font> &amp; <font color="Maroon" title="c9">E</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c1">F</font> implies <font color="Maroon" title="c9">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a> <font color="Maroon" title="c1">F</font> )</span><br/><span class="kw">assume </span><a NAME="E1:137_1_1"/><span class="lab"><font color="Green" title="E68">F1</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="c9">E</font> &amp; <font color="Maroon" title="c9">E</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c1">F</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a> <font color="Maroon" title="c1">F</font></span><br/><a NAME="E2:137_1_1"/>
<font color="Maroon" title="c1">F</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T7" onmouseover="rs('field_4/T7')" onmouseout="rh()">FIELD_4:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:137_1_1"/>
<font color="Maroon" title="c9">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:137_1_1"><span class="lab"><font color="Green" title="E68">F1</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="E10:137_1"/>
<font color="Maroon" title="c1">F</font> 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="#E7:137_1"><span class="lab"><font color="Green" title="E66">E</font></span></a>, <a class="txt" href="#E8:137_1"><span class="lab"><font color="Green" title="E67">F</font></span></a>, <a class="txt" href="#E2:137"><span class="lab"><font color="Green" title="E62">A</font></span></a>, <a class="txt" href="#E5:137_1"><span class="lab"><font color="Green" title="E65">D</font></span></a>, <a class="ref" href="field_4.html#D5" onmouseover="rs('field_4/D5')" onmouseout="rh()">FIELD_4:def 5</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/></div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="137_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"> ( <font color="Maroon" title="c1">F</font> is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of <font color="Maroon" title="c2">p</font> implies  <a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:137_2"/>
<font color="Maroon" title="c1">F</font> is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a> </span><br/><a NAME="E2:137_2"/><span class="kw">then </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="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_8.html#D1" onmouseover="rs('field_8/D1')" onmouseout="rh()">FIELD_8:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</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="c7">q</font> being    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E4:137_2"/><span class="lab"><font color="Green" title="E64">G</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">x</font> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <font color="Maroon" title="c7">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D5" onmouseover="rs('field_4/D5')" onmouseout="rh()">FIELD_4:def 5</a></span>;<br/><a NAME="E5:137_2"/>
 <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:137_2"><span class="lab"><font color="Green" title="E64">G</font></span></a></span>;<br/><a NAME="E6:137_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c</font></span>)</span> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:137"><span class="lab"><font color="Green" title="E62">A</font></span></a>, <a class="ref" href="field_9.html#T26" target="_self" onmouseover="rs('field_9/T26')" onmouseout="rh()">lemeval2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:137_2"/>
 <a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:137"><span class="lab"><font color="Green" title="E62">A</font></span></a>, <a class="ref" href="field_9.html#D9" target="_self" onmouseover="rs('field_9/D9')" onmouseout="rh()">defDC</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:137"/>
( <font color="Maroon" title="c1">F</font> is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of <font color="Maroon" title="c2">p</font> iff  <a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <font color="Maroon" title="c2">p</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:137"><span class="lab"><font color="Green" title="E63">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/>


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