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

<span class="kw">set </span><font color="Maroon" title="c3">F</font> =  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">FX</font> =  <a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p1">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f</font> =  <a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">Kr</font> =  <a href="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">p</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">o</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1: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="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:48_1_1"/><span class="lab"><font color="Green" title="E20">X</font></span>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> <font color="Maroon" title="c1">n</font></span>)</span> <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> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<span class="p2">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">p</font>)</span>)</span> <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="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<span class="p2">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">p</font>)</span>)</span> )
 
;<br/>
<span class="lab"><font color="Green" title="E21">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_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> = 
 <a href="field_2.html#K2" title="FIELD_2:func.2">carr</a> <span class="p1">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_2.html#D7" onmouseover="rs('field_2/D7')" onmouseout="rh()">FIELD_2:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<span class="p3">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p3">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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> <font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1"><span class="lab"><font color="Green" title="E20">X</font></span></a>, <a class="ref" href="field_2.html#D2" onmouseover="rs('field_2/D2')" onmouseout="rh()">FIELD_2:def 2</a></span>
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_1_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">  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1_1_2"/>
 not  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</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="E2:48_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</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_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_3.html#D3" onmouseover="rs('field_3/D3')" onmouseout="rh()">FIELD_3:def 3</a></span>;<br/><a NAME="E3:48_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A</font></span>: 
(  the    <a href="subset_1.html#M1" title="SUBSET_1: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="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> &amp;  the    <a href="subset_1.html#M1" title="SUBSET_1: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="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">q</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1: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="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> as    <a href="ring_4.html#M1" title="RING_4:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:48_1_1_2"/>
 <a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K3" title="GR_CY_1:func.3">addint</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#D12" onmouseover="rs('int_3/D12')" onmouseout="rh()">INT_3:def 12</a></span>;<br/><a NAME="E6:48_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_3.html#T24" onmouseover="rs('field_3/T24')" onmouseout="rh()">FIELD_3:24</a></span>;<br/><a NAME="E7:48_1_1_2"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1: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="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<span class="p2">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E21">H</font></span></a>, <a class="txt" href="#E3:48_1_1_2"><span class="lab"><font color="Green" title="E23">A</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">o</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1: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="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p3">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a></span>)</span></span>)</span> ;<br/><span class="kw">consider </span><font color="Maroon" title="c10">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E10:48_1_1_2"/><span class="lab"><font color="Green" title="E24">B</font></span>: 
<font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> (<span class="p2">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p3">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c10">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_1.html#T11" onmouseover="rs('ring_1/T11')" onmouseout="rh()">RING_1:11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> ;<br/><a NAME="E12:48_1_1_2"/>
<font color="Maroon" title="c10">a</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c10">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="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> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T15" onmouseover="rs('rlvect_1/T15')" onmouseout="rh()">RLVECT_1:15</a></span>;<br/><a NAME="E13:48_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">D0</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_1_2"><span class="lab"><font color="Green" title="E24">B</font></span></a>, <a class="ref" href="ring_1.html#T5" onmouseover="rs('ring_1/T5')" onmouseout="rh()">RING_1:5</a>, <a class="ref" href="ideal_1.html#T3" onmouseover="rs('ideal_1/T3')" onmouseout="rh()">IDEAL_1:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E15:48_1_1_2"/><span class="lab"><font color="Green" title="E26">D1</font></span>: 
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <font color="Maroon" title="c13">z</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_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/><a NAME="E16:48_1_1_2"/><span class="lab"><font color="Green" title="E27">D2</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_1_2"><span class="lab"><font color="Green" title="E25">D0</font></span></a>, <a class="txt" href="#E15:48_1_1_2"><span class="lab"><font color="Green" title="E26">D1</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c12">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_1_1_2"><span class="lab"><font color="Green" title="E26">D1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">a</font> = <font color="Maroon" title="c10">a</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> <font color="Maroon" title="c1">n</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="E19:48_1_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E20:48_1_1_2"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">a</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/><a NAME="E21:48_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">E</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_1_1_2"><span class="lab"><font color="Green" title="E26">D1</font></span></a>, <a class="txt" href="#E16:48_1_1_2"><span class="lab"><font color="Green" title="E27">D2</font></span></a>, <a class="ref" href="tarski.html#D5" onmouseover="rs('tarski/D5')" onmouseout="rh()">TARSKI:def 5</a></span>;<br/><div><span class="lab"><font color="Green" title="E29">F</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_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 <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1_1_2_1"/><span class="lab"><font color="Green" title="E30">E3</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI: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"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_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="E1:48_1_1_2_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1_1_2"><span class="lab"><font color="Green" title="E28">E</font></span></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="48_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">i</font></span><a href="tarski.html#K1" title="TARSKI: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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">F1</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D5" onmouseover="rs('tarski/D5')" onmouseout="rh()">TARSKI:def 5</a></span>;<br/><a NAME="E3:48_1_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E4:48_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">F3</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_2_1"><span class="lab"><font color="Green" title="E30">E3</font></span></a>, <a class="txt" href="#E2:48_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E31">F1</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E5:48_1_1_2_1_1_1"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E32">F3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI: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" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">F1</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D5" onmouseover="rs('tarski/D5')" onmouseout="rh()">TARSKI:def 5</a></span>;<br/><a NAME="E3:48_1_1_2_1_1_2"/>
<font color="Maroon" title="c14">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_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="E1:48_1_1_2_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E31">F1</font></span></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="48_1_1_2_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E32">F3</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI: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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_1_1_2_1_1"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_1_2_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E32">F3</font></span></a>, <a class="txt" href="#E1:48_1_1_2_1"><span class="lab"><font color="Green" title="E30">E3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI: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" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E32">F3</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_1_1_2_1_2"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_1_2_1_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E32">F3</font></span></a>, <a class="txt" href="#E1:48_1_1_2_1"><span class="lab"><font color="Green" title="E30">E3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI: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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_2"><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="E1:48_1_1_2_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1_1_2"><span class="lab"><font color="Green" title="E28">E</font></span></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="48_1_1_2_2_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">i</font></span><a href="tarski.html#K1" title="TARSKI: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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">E1</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D5" onmouseover="rs('tarski/D5')" onmouseout="rh()">TARSKI:def 5</a></span>;<br/><a NAME="E3:48_1_1_2_2_1"/>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_1_1_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:48_1_1_2"><span class="lab"><font color="Green" title="E29">F</font></span></a>, <a class="txt" href="#E2:48_1_1_2_2_1"><span class="lab"><font color="Green" title="E30">E1</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI: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" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_2_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">E1</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D5" onmouseover="rs('tarski/D5')" onmouseout="rh()">TARSKI:def 5</a></span>;<br/><a NAME="E3:48_1_1_2_2_2"/>
<font color="Maroon" title="c14">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E4:48_1_1_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">F0</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:48_1_1_2"><span class="lab"><font color="Green" title="E29">F</font></span></a>, <a class="txt" href="#E2:48_1_1_2_2_2"><span class="lab"><font color="Green" title="E30">E1</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_2_2_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="E1:48_1_1_2_2_2_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 or 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_2_2_1_1"/><span class="lab"><font color="Green" title="E32">E3</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1
 ; <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"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_2_2_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="E1:48_1_1_2_2_2_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1_1_2"><span class="lab"><font color="Green" title="E28">E</font></span></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="48_1_1_2_2_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_2_2_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">i</font></span><a href="tarski.html#K1" title="TARSKI: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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_2_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">F1</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D5" onmouseover="rs('tarski/D5')" onmouseout="rh()">TARSKI:def 5</a></span>;<br/><a NAME="E3:48_1_1_2_2_2_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">F3</font></span>: <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
<font color="Maroon" title="c14">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E33">F1</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_2_2_2_1_1"><span class="lab"><font color="Green" title="E32">E3</font></span></a>, <a class="txt" href="#E4:48_1_1_2_2_2"><span class="lab"><font color="Green" title="E31">F0</font></span></a>, <a class="ref" href="enumset1.html#T29" onmouseover="rs('enumset1/T29')" onmouseout="rh()">ENUMSET1:29</a></span>
;<br/><a NAME="E5:48_1_1_2_2_2_1_1_1_1"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_1_1_2_2_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_2_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E34">F3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI: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" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_2_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_2_2_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_2_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">F1</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D5" onmouseover="rs('tarski/D5')" onmouseout="rh()">TARSKI:def 5</a></span>;<br/><a NAME="E3:48_1_1_2_2_2_1_1_1_2"/>
<font color="Maroon" title="c14">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_2_2_1_1_1_2_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="E1:48_1_1_2_2_2_1_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E33">F1</font></span></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="48_1_1_2_2_2_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_2_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI: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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_2_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">F3</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_2_2_2_1_1"><span class="lab"><font color="Green" title="E32">E3</font></span></a>, <a class="txt" href="#E4:48_1_1_2_2_2"><span class="lab"><font color="Green" title="E31">F0</font></span></a>, <a class="ref" href="enumset1.html#T29" onmouseover="rs('enumset1/T29')" onmouseout="rh()">ENUMSET1:29</a></span>;<br/><a NAME="E3:48_1_1_2_2_2_1_1_1_2_1_1"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_1_1_2_2_2_1_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_2_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">F3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI: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" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_2_2_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_2_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_2_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">F3</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_2_2_2_1_1"><span class="lab"><font color="Green" title="E32">E3</font></span></a>, <a class="txt" href="#E4:48_1_1_2_2_2"><span class="lab"><font color="Green" title="E31">F0</font></span></a>, <a class="ref" href="enumset1.html#T29" onmouseover="rs('enumset1/T29')" onmouseout="rh()">ENUMSET1:29</a></span>;<br/><a NAME="E3:48_1_1_2_2_2_1_1_1_2_1_2"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_1_1_2_2_2_1_1_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_2_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E34">F3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI: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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_2_2_1_2"/>
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1
 ; <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"><a NAME="E2:48_1_1_2_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">G1</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_2_2_2"><span class="lab"><font color="Green" title="E31">F0</font></span></a>, <a class="ref" href="field_3.html#T2" onmouseover="rs('field_3/T2')" onmouseout="rh()">FIELD_3:2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_2_2_1_2_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="E1:48_1_1_2_2_2_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1_1_2"><span class="lab"><font color="Green" title="E28">E</font></span></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="48_1_1_2_2_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_2_2_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">i</font></span><a href="tarski.html#K1" title="TARSKI: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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_2_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">F1</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_2_2_2"><span class="lab"><font color="Green" title="E31">F0</font></span></a>, <a class="ref" href="tarski.html#D5" onmouseover="rs('tarski/D5')" onmouseout="rh()">TARSKI:def 5</a></span>;<br/><a NAME="E3:48_1_1_2_2_2_1_2_1_1"/>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_1_1_2_2_2_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_2_2_1_2"><span class="lab"><font color="Green" title="E32">G1</font></span></a>, <a class="txt" href="#E2:48_1_1_2_2_2_1_2_1_1"><span class="lab"><font color="Green" title="E33">F1</font></span></a>, <a class="ref" href="card_1.html#T50" onmouseover="rs('card_1/T50')" onmouseout="rh()">CARD_1:50</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="48_1_1_2_2_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_2_2_1_2_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_2_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">F1</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_2_2_2"><span class="lab"><font color="Green" title="E31">F0</font></span></a>, <a class="txt" href="#E2:48_1_1_2_2_2_1_2"><span class="lab"><font color="Green" title="E32">G1</font></span></a>, <a class="ref" href="tarski.html#D5" onmouseover="rs('tarski/D5')" onmouseout="rh()">TARSKI:def 5</a>, <a class="ref" href="card_1.html#T50" onmouseover="rs('card_1/T50')" onmouseout="rh()">CARD_1:50</a></span>;<br/><a NAME="E3:48_1_1_2_2_2_1_2_1_2"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_2_2_1_2_1_2_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="E1:48_1_1_2_2_2_1_2_1_2_1"/><span class="kw">then </span>
( 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_2_2_1_2_1_2"><span class="lab"><font color="Green" title="E33">F1</font></span></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="48_1_1_2_2_2_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_2_2_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E34">F2</font></span>: 
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI: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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_2_2_1_2_1_2_1_1"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_1_2_2_2_1_2_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_2_2_2_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E34">F2</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="48_1_1_2_2_2_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_2_2_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E34">F2</font></span>: 
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.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"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_2_2_1_2_1_2_1_2"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_1_2_2_2_1_2_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_2_2_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E34">F2</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></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="E4:48_1_1"/>
 <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</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/>


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