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

<span class="kw">let </span><font color="Maroon" title="c1">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> <span class="p2">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> implies  <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2 is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of <font color="Maroon" title="c1">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:92"/><span class="lab"><font color="Green" title="E47">A</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</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">  <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2 is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of <font color="Maroon" title="c1">p</font></span><br/>

<a NAME="E2:92"/><span class="lab"><font color="Green" title="E48">B</font></span>: 
 <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2 is   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of  <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2
 
<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="92_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  <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2  st <font color="Maroon" title="c1">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  <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2 holds <br/><font color="Olive" title="b1">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">E</font> be   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of  <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2; <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">p</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Maroon" title="c2">E</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of  <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2 implies <font color="Maroon" title="c2">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2 )</span><br/><span class="kw">assume </span><a NAME="E1:92_1"/><span class="lab"><font color="Green" title="E49">C</font></span>: 
( <font color="Maroon" title="c1">p</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Maroon" title="c2">E</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of  <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2 )
 ; <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">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span><br/><a NAME="E2:92_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">D</font></span>: 
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</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 <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span> &amp;  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font> &amp;  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font> &amp;  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#D1" onmouseover="rs('ec_pf_1/D1')" onmouseout="rh()">EC_PF_1:def 1</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="92_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">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">o</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="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span> holds <br/><font color="Olive" title="b1">o</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 <font color="Maroon" title="c2">E</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">o</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">o</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="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span> implies <font color="Olive">b<sub>1</sub></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 <font color="Maroon" title="c2">E</font> )</span><br/><span class="kw">assume </span><a NAME="E1:92_1_1"/>
<font color="Maroon" title="c3">o</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="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</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="Olive">b<sub>1</sub></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 <font color="Maroon" title="c2">E</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:92_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span> or <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T34" target="_self" onmouseover="rs('field_9/T34')" onmouseout="rh()">cz2</a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:92_1_1_1_1"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</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="Olive">b<sub>1</sub></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 <font color="Maroon" title="c2">E</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:92_1_1_1_1"/>
<font color="Maroon" title="c3">o</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 <font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1"><span class="lab"><font color="Green" title="E50">D</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:92_1_1_1_2"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</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="Olive">b<sub>1</sub></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 <font color="Maroon" title="c2">E</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:92_1_1_1_2"/>
<font color="Maroon" title="c3">o</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 <font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1"><span class="lab"><font color="Green" title="E50">D</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E4:92_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1"><span class="lab"><font color="Green" title="E50">D</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>;<br/><a NAME="E5:92_1"/><span class="kw">then </span>
 <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2 is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1"><span class="lab"><font color="Green" title="E50">D</font></span></a>, <a class="ref" href="ec_pf_1.html#D1" onmouseover="rs('ec_pf_1/D1')" onmouseout="rh()">EC_PF_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:92_1"/>
<font color="Maroon" title="c2">E</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a>  <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1"><span class="lab"><font color="Green" title="E49">C</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="E4:92"/>
 <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2 is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E47">A</font></span></a>, <a class="txt" href="field_9.html#E66"><span class="lab"><font color="Green" title="E45">z25</font></span></a>, <a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E48">B</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/>


</div>
