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

<span class="kw">set </span><font color="Maroon" title="c1">p</font> = <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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">a</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</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>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="89_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="polynom5.html#K7" title="POLYNOM5:func.7">Roots</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 href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:89_1"/><span class="lab"><font color="Green" title="E45">A</font></span>: 
 <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</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 href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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 NAME="E2:89_1"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</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 href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</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>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">a</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</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> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span> ;<br/><a NAME="E4:89_1"/><span class="lab"><font color="Green" title="E46">B</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89_1"><span class="lab"><font color="Green" title="E45">A</font></span></a>, <a class="ref" href="polynom5.html#D10" onmouseover="rs('polynom5/D10')" onmouseout="rh()">POLYNOM5:def 10</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_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="E1:89_1_1"/>
( <font color="Maroon" title="c3">a</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">a</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="89_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:89_1_1_1"/>
<font color="Maroon" title="c3">a</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"> contradiction</span><br/><div class="add"><span class="kw">then </span> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c3">a</font>) = 
<span class="p1">(<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> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</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"><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> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T20" target="_self" onmouseover="rs('field_9/T20')" onmouseout="rh()">evalq</a></span>
<br/>.= 
 <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>

;<br/><span class="kw">hence </span><a NAME="E3:89_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89_1"><span class="lab"><font color="Green" title="E46">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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:89_1_1_2"/>
<font color="Maroon" title="c3">a</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"> contradiction</span><br/><div class="add"><span class="kw">then </span> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c3">a</font>) = 
<span class="p1">(<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> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</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"><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> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T20" target="_self" onmouseover="rs('field_9/T20')" onmouseout="rh()">evalq</a></span>
<br/>.= 
<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="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <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="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_3.html#T4" onmouseover="rs('field_3/T4')" onmouseout="rh()">FIELD_3:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:89_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89_1"><span class="lab"><font color="Green" title="E46">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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:89"/>
 <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <a href="field_9.html#K11" title="FIELD_9:func.11">X^2+X+1</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"> verum</span><br/>


</div>
