<?xml version="1.0"?>
<div class="add">

<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">a</font> being   non  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font><br/>  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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> <font color="Olive" title="b1">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> iff  ex <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">z</font> being  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> <font color="Olive" title="b1">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> st <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> <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#K1" title="FIELD_9:func.1">@</a> <span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b4">z</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   non  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> iff  ex <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <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#K1" title="FIELD_9:func.1">@</a> <span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> iff  ex <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">z</font> being  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> <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#K1" title="FIELD_9:func.1">@</a> <span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">z</font></span>)</span> )</span><br/>





<a NAME="E1:128"/><span class="lab"><font color="Green" title="E61">H</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_6.html#K3" title="FIELD_6:func.3">RAdj</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> <font color="Maroon" title="c2">a</font></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> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Olive" title="b1">p</font>,<span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T45" onmouseover="rs('field_6/T45')" onmouseout="rh()">FIELD_6:45</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_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"> ( <font color="Maroon" title="c3">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">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> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) implies  ex <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">z</font> being  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> <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#K1" title="FIELD_9:func.1">@</a> <span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">z</font></span>)</span> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:128_1"/>
<font color="Maroon" title="c3">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">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> <font color="Maroon" title="c2">a</font></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">  ex <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">z</font> being  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> <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#K1" title="FIELD_9:func.1">@</a> <span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">z</font></span>)</span></span><br/><a NAME="E2:128_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Olive" title="b1">p</font>,<span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E61">H</font></span></a>, <a class="ref" href="field_6.html#T56" onmouseover="rs('field_6/T56')" onmouseout="rh()">FIELD_6:56</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">p1</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E4:128_1"/><span class="lab"><font color="Green" title="E62">A</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c4">p1</font>,<span class="p1">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)
 ;<br/><span class="kw">set </span><font color="Maroon" title="c5">ma</font> =  <a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p1">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>);<br/><span class="kw">set </span><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <span class="p1">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span>;<br/><a NAME="E5:128_1"/><span class="lab"><font color="Green" title="E63">B</font></span>: 
<font color="Maroon" title="c1">F</font> is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <span class="p2">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D1" onmouseover="rs('field_4/D1')" onmouseout="rh()">FIELD_4:def 1</a></span>;<br/><a NAME="E6:128_1"/><span class="lab"><font color="Green" title="E64">C</font></span>: 
<font color="Maroon" title="c4">p1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <span class="p3">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p4">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <span class="p2">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#T4" onmouseover="rs('ring_4/T4')" onmouseout="rh()">RING_4:4</a></span>;<br/><span class="lab"><font color="Green" title="E65">D</font></span>:  <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c4">p1</font>,<span class="p1">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>) = 
<span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <span class="p4">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p5">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p3">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p4">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <span class="p3">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p4">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:128_1"><span class="lab"><font color="Green" title="E64">C</font></span></a>, <a class="txt" href="#E5:128_1"><span class="lab"><font color="Green" title="E63">B</font></span></a>, <a class="ref" href="algnum_1.html#T15" onmouseover="rs('algnum_1/T15')" onmouseout="rh()">ALGNUM_1:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <span class="p4">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p5">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p3">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p4">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <span class="p3">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p4">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_1"><span class="lab"><font color="Green" title="E63">B</font></span></a>, <a class="ref" href="algnum_1.html#T20" onmouseover="rs('algnum_1/T20')" onmouseout="rh()">ALGNUM_1:20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <span class="p4">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p5">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <span class="p5">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <span class="p3">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p4">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T52" onmouseover="rs('field_6/T52')" onmouseout="rh()">FIELD_6:52</a></span>
<br/>.= 
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <span class="p2">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)

;<br/><a NAME="E8:128_1"/>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <span class="p2">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p1">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#T16" onmouseover="rs('field_5/T16')" onmouseout="rh()">FIELD_5:16</a></span>;<br/><a NAME="E9:128_1"/><span class="kw">then </span>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <span class="p2">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p1">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="field_9.html#E83"><span class="lab"><font color="Green" title="E52">m30</font></span></a></span>;<br/><a NAME="E10:128_1"/><span class="kw">then </span>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <span class="p2">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T18" target="_self" onmouseover="rs('field_9/T18')" onmouseout="rh()">qua4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> being  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E12:128_1"/><span class="lab"><font color="Green" title="E66">E</font></span>: 
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <span class="p2">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="field_9.html#K1" title="FIELD_9:func.1">@</a> <span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> <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#K1" title="FIELD_9:func.1">@</a> <span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T15" target="_self" onmouseover="rs('field_9/T15')" onmouseout="rh()">deg2evale</a></span>;<br/><a NAME="E13:128_1"/>
<font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <span class="p1">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span> is    <a href="ring_4.html#M1" title="RING_4:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D10" onmouseover="rs('polynom3/D10')" onmouseout="rh()">POLYNOM3:def 10</a></span>;<br/><a NAME="E14:128_1"/><span class="kw">then </span>
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <span class="p2">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="field_9.html#K1" title="FIELD_9:func.1">@</a> <span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128_1"><span class="lab"><font color="Green" title="E62">A</font></span></a>, <a class="txt" href="#E7:128_1"><span class="lab"><font color="Green" title="E65">D</font></span></a>, <a class="ref" href="field_7.html#T14" onmouseover="rs('field_7/T14')" onmouseout="rh()">FIELD_7:14</a></span>;<br/><span class="kw">hence </span><a NAME="E15:128_1"/>
 ex <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">z</font> being  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> <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#K1" title="FIELD_9:func.1">@</a> <span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:128_1"><span class="lab"><font color="Green" title="E66">E</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>
<span class="kw">hence </span><a NAME="E3:128"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> iff  ex <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">z</font> being  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> <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#K1" title="FIELD_9:func.1">@</a> <span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">z</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"> verum</span><br/>


</div>
