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

<b>set </b><font color="Maroon" title="c1">Z</font> = <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c2">N</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>;<br/>
<a NAME="E1:12_2_1"/><i><font color="Green" title="E9">A1</font></i>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D10" title="LATTICE3:def.10">LATTICE3:def 10</a><br/>

<a NAME="E1:12_2_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
<span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></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="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<b>thus </b><a NAME="E2:12_2_1_1"/>
 ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> st <br/>( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; ( for <font color="Olive" title="b2">z'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>  st <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">z'</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">z'</font> holds <br/><font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">z'</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1_1_1_1"/>
( ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 </span><b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_1"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_2"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_2"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_2"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_2"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_3"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_3"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_3"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_3"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_3"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_3"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_4"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_4"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_4"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_4"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_4"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_4"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_5"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_5"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_5"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_5"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_5"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_5"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_5"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_6"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_6"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_6"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_6"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_6"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_6"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_6"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_6"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_7"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_7"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_7"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_7"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_7"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_7"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_7"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_7"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_8"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_8"/><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 )
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_1_1_8"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">z</font> = 3 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E4:12_2_1_1_1_1_8"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_8_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_2_1_1_1_1_8_1"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2_1_1_1_1_8_1_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 ;<br/>

<a NAME="E2:12_2_1_1_1_1_8_1_1"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_8"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_1_1_1_1_8_1_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:12_2_1_1_1_1_8_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 ;<br/>

<a NAME="E3:12_2_1_1_1_1_8_1"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_8"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:12_2_1_1_1_1_8_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:12_2_1_1_1_1_8"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><a NAME="E6:12_2_1_1_1_1_8"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c6">z'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><b>assume </b><a NAME="E7:12_2_1_1_1_1_8"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font> )
 ;<br/><a NAME="E8:12_2_1_1_1_1_8"/><b>then </b><i><font color="Green" title="E13">A5</font></i>: 
( 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z'</font> &amp; 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z'</font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_8"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><div><i><font color="Green" title="E14">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_8_2"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:12_2_1_1_1_1_8_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_8_2"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_8"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_8_2"><i><font color="Green" title="E15">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E15">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_8_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_8_3"/><i><font color="Green" title="E16">A9</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:12_2_1_1_1_1_8_3"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:12_2_1_1_1_1_8_3"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">z'</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_8"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_8_3"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_8_3"><i><font color="Green" title="E16">A9</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E16">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_8_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_8_4"/><i><font color="Green" title="E17">A11</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ;<br/><a NAME="E2:12_2_1_1_1_1_8_4"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_8_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_8"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_8_4"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_8_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_8_5"/><i><font color="Green" title="E17">A12</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 ;<br/><a NAME="E2:12_2_1_1_1_1_8_5"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_8_5"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_8"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_8_5"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self" title="YELLOW11:th.4">Th4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:12_2_1_1_1_1_8"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E6:12_2_1_1_1_1_8"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="yellow11.html#E9:12_2_1_1_1_1_8"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="yellow11.html#E10:12_2_1_1_1_1_8"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="yellow11.html#E11:12_2_1_1_1_1_8"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_9"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_9"/><i><font color="Green" title="E11">A13</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 )
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_1_1_9"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">z</font> = 3 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E4:12_2_1_1_1_1_9"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_9_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_2_1_1_1_1_9_1"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_9_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2_1_1_1_1_9_1_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 ;<br/>

<a NAME="E2:12_2_1_1_1_1_9_1_1"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_9"><i><font color="Green" title="E11">A13</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_1_1_1_1_9_1_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:12_2_1_1_1_1_9_1"/>
<font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_9"><i><font color="Green" title="E11">A13</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:12_2_1_1_1_1_9"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><a NAME="E6:12_2_1_1_1_1_9"/><i><font color="Green" title="E12">A14</font></i>: 
<font color="Maroon" title="c6">z'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><b>assume </b><a NAME="E7:12_2_1_1_1_1_9"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font> )
 ;<br/><a NAME="E8:12_2_1_1_1_1_9"/><b>then </b><i><font color="Green" title="E13">A15</font></i>: 
( 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z'</font> &amp; 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z'</font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_9"><i><font color="Green" title="E11">A13</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><div><i><font color="Green" title="E14">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_9_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_9_2"/><i><font color="Green" title="E15">A17</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:12_2_1_1_1_1_9_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_9_2"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_9"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_9_2"><i><font color="Green" title="E15">A17</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E15">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_9_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_9_3"/><i><font color="Green" title="E16">A19</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:12_2_1_1_1_1_9_3"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 <b>by </b><i><a class="ref" href="yellow11.html#T4" target="_self" title="YELLOW11:th.4">Th4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_9_3"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_9"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_9_3"><i><font color="Green" title="E16">A19</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E16">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_9_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_9_4"/><i><font color="Green" title="E17">A21</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ;<br/><a NAME="E2:12_2_1_1_1_1_9_4"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_9_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_9"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_9_4"><i><font color="Green" title="E17">A21</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_9_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_9_5"/><i><font color="Green" title="E17">A22</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 ;<br/><a NAME="E2:12_2_1_1_1_1_9_5"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_9_5"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_9"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_9_5"><i><font color="Green" title="E17">A22</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self" title="YELLOW11:th.4">Th4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:12_2_1_1_1_1_9"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E6:12_2_1_1_1_1_9"><i><font color="Green" title="E12">A14</font></i></a>, <a class="txt" href="yellow11.html#E9:12_2_1_1_1_1_9"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="yellow11.html#E10:12_2_1_1_1_1_9"><i><font color="Green" title="E15">A18</font></i></a>, <a class="txt" href="yellow11.html#E11:12_2_1_1_1_1_9"><i><font color="Green" title="E16">A20</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_10"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_10"/><i><font color="Green" title="E11">A23</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_1_1_10"/>
1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 3
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2_1_1_1_1_10_1"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 ;<br/>

<a NAME="E2:12_2_1_1_1_1_10_1"/><b>then </b>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_1_1_1_1_10_1"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3
 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_10"><i><font color="Green" title="E11">A23</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E4:12_2_1_1_1_1_10"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E5:12_2_1_1_1_1_10"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E6:12_2_1_1_1_1_10"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E7:12_2_1_1_1_1_10"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E8:12_2_1_1_1_1_10"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E9:12_2_1_1_1_1_10"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_11"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_11"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_11"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_11"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_11"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_11"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_11"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_11"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_12"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_12"/><i><font color="Green" title="E11">A24</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_1_1_12"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">z</font> = 3 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E4:12_2_1_1_1_1_12"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_12_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_2_1_1_1_1_12_1"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_12_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2_1_1_1_1_12_1_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 ;<br/>

<a NAME="E2:12_2_1_1_1_1_12_1_1"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_12"><i><font color="Green" title="E11">A24</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_1_1_1_1_12_1_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:12_2_1_1_1_1_12_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 ;<br/>

<a NAME="E3:12_2_1_1_1_1_12_1"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_12"><i><font color="Green" title="E11">A24</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:12_2_1_1_1_1_12_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:12_2_1_1_1_1_12"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><a NAME="E6:12_2_1_1_1_1_12"/><i><font color="Green" title="E12">A25</font></i>: 
<font color="Maroon" title="c6">z'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><b>assume </b><a NAME="E7:12_2_1_1_1_1_12"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font> )
 ;<br/><a NAME="E8:12_2_1_1_1_1_12"/><b>then </b><i><font color="Green" title="E13">A26</font></i>: 
( 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z'</font> &amp; 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z'</font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_12"><i><font color="Green" title="E11">A24</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><div><i><font color="Green" title="E14">A27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_12_2"/><i><font color="Green" title="E15">A28</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:12_2_1_1_1_1_12_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_12_2"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_12"><i><font color="Green" title="E13">A26</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_12_2"><i><font color="Green" title="E15">A28</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E15">A29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_12_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_12_3"/><i><font color="Green" title="E16">A30</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:12_2_1_1_1_1_12_3"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:12_2_1_1_1_1_12_3"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">z'</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_12"><i><font color="Green" title="E13">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_12_3"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_12_3"><i><font color="Green" title="E16">A30</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E16">A31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_12_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_12_4"/><i><font color="Green" title="E17">A32</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ;<br/><a NAME="E2:12_2_1_1_1_1_12_4"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_12_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_12"><i><font color="Green" title="E13">A26</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_12_4"><i><font color="Green" title="E17">A32</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_12_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_12_5"/><i><font color="Green" title="E17">A33</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 ;<br/><a NAME="E2:12_2_1_1_1_1_12_5"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_12_5"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_12"><i><font color="Green" title="E13">A26</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_12_5"><i><font color="Green" title="E17">A33</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self" title="YELLOW11:th.4">Th4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:12_2_1_1_1_1_12"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E6:12_2_1_1_1_1_12"><i><font color="Green" title="E12">A25</font></i></a>, <a class="txt" href="yellow11.html#E9:12_2_1_1_1_1_12"><i><font color="Green" title="E14">A27</font></i></a>, <a class="txt" href="yellow11.html#E10:12_2_1_1_1_1_12"><i><font color="Green" title="E15">A29</font></i></a>, <a class="txt" href="yellow11.html#E11:12_2_1_1_1_1_12"><i><font color="Green" title="E16">A31</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_13"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_13"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_13"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_13"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_13"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_13"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_13"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_13"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_14"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_14"/><i><font color="Green" title="E11">A34</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 )
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_1_1_14"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">z</font> = 3 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E4:12_2_1_1_1_1_14"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_14_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_2_1_1_1_1_14_1"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_14_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2_1_1_1_1_14_1_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 ;<br/>

<a NAME="E2:12_2_1_1_1_1_14_1_1"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_14"><i><font color="Green" title="E11">A34</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_1_1_1_1_14_1_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:12_2_1_1_1_1_14_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 ;<br/>

<b>hence </b><a NAME="E3:12_2_1_1_1_1_14_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_14"><i><font color="Green" title="E11">A34</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:12_2_1_1_1_1_14"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><a NAME="E6:12_2_1_1_1_1_14"/><i><font color="Green" title="E12">A35</font></i>: 
<font color="Maroon" title="c6">z'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><b>assume </b><a NAME="E7:12_2_1_1_1_1_14"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font> )
 ;<br/><a NAME="E8:12_2_1_1_1_1_14"/><b>then </b><i><font color="Green" title="E13">A36</font></i>: 
( 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z'</font> &amp; 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z'</font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_14"><i><font color="Green" title="E11">A34</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><div><i><font color="Green" title="E14">A37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_14_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_14_2"/><i><font color="Green" title="E15">A38</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:12_2_1_1_1_1_14_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_14_2"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_14"><i><font color="Green" title="E13">A36</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_14_2"><i><font color="Green" title="E15">A38</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E15">A39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_14_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_14_3"/><i><font color="Green" title="E16">A40</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:12_2_1_1_1_1_14_3"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:12_2_1_1_1_1_14_3"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">z'</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_14"><i><font color="Green" title="E13">A36</font></i></a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_14_3"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_14_3"><i><font color="Green" title="E16">A40</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E16">A41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_14_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_14_4"/><i><font color="Green" title="E17">A42</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ;<br/><a NAME="E2:12_2_1_1_1_1_14_4"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 <b>by </b><i><a class="ref" href="yellow11.html#T4" target="_self" title="YELLOW11:th.4">Th4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_14_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_14"><i><font color="Green" title="E13">A36</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_14_4"><i><font color="Green" title="E17">A42</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_14_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_14_5"/><i><font color="Green" title="E17">A43</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 ;<br/><a NAME="E2:12_2_1_1_1_1_14_5"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_14_5"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_14"><i><font color="Green" title="E13">A36</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_14_5"><i><font color="Green" title="E17">A43</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self" title="YELLOW11:th.4">Th4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:12_2_1_1_1_1_14"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E6:12_2_1_1_1_1_14"><i><font color="Green" title="E12">A35</font></i></a>, <a class="txt" href="yellow11.html#E9:12_2_1_1_1_1_14"><i><font color="Green" title="E14">A37</font></i></a>, <a class="txt" href="yellow11.html#E10:12_2_1_1_1_1_14"><i><font color="Green" title="E15">A39</font></i></a>, <a class="txt" href="yellow11.html#E11:12_2_1_1_1_1_14"><i><font color="Green" title="E16">A41</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_15"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_15"/><i><font color="Green" title="E11">A44</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_1_1_15"/>
2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 3
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2_1_1_1_1_15_1"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ;<br/>

<a NAME="E2:12_2_1_1_1_1_15_1"/><b>then </b>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_1_1_1_1_15_1"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3
 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_15"><i><font color="Green" title="E11">A44</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E4:12_2_1_1_1_1_15"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E5:12_2_1_1_1_1_15"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E6:12_2_1_1_1_1_15"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E7:12_2_1_1_1_1_15"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E8:12_2_1_1_1_1_15"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E9:12_2_1_1_1_1_15"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_16"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_16"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_16"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_16"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_16"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_16"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_16"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_16"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_17"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_17"/><i><font color="Green" title="E11">A45</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_1_1_17"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">z</font> = 3 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E4:12_2_1_1_1_1_17"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_17_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_2_1_1_1_1_17_1"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_17"><i><font color="Green" title="E11">A45</font></i></a></i>;<br/>

<b>thus </b><a NAME="E2:12_2_1_1_1_1_17_1"/>
<font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_17_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2_1_1_1_1_17_1_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 ;<br/>

<a NAME="E2:12_2_1_1_1_1_17_1_1"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_17"><i><font color="Green" title="E11">A45</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_1_1_1_1_17_1_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E5:12_2_1_1_1_1_17"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><a NAME="E6:12_2_1_1_1_1_17"/><i><font color="Green" title="E12">A46</font></i>: 
<font color="Maroon" title="c6">z'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><b>assume </b><a NAME="E7:12_2_1_1_1_1_17"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font> )
 ;<br/><a NAME="E8:12_2_1_1_1_1_17"/><b>then </b><i><font color="Green" title="E13">A47</font></i>: 
( 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z'</font> &amp; 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z'</font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_17"><i><font color="Green" title="E11">A45</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><div><i><font color="Green" title="E14">A48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_17_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_17_2"/><i><font color="Green" title="E15">A49</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:12_2_1_1_1_1_17_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_17_2"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_17"><i><font color="Green" title="E13">A47</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_17_2"><i><font color="Green" title="E15">A49</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E15">A50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_17_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_17_3"/><i><font color="Green" title="E16">A51</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:12_2_1_1_1_1_17_3"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 <b>by </b><i><a class="ref" href="yellow11.html#T4" target="_self" title="YELLOW11:th.4">Th4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_17_3"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_17"><i><font color="Green" title="E13">A47</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_17_3"><i><font color="Green" title="E16">A51</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E16">A52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_17_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_17_4"/><i><font color="Green" title="E17">A53</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ;<br/><a NAME="E2:12_2_1_1_1_1_17_4"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_17_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_17"><i><font color="Green" title="E13">A47</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_17_4"><i><font color="Green" title="E17">A53</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_17_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_17_5"/><i><font color="Green" title="E17">A54</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 ;<br/><a NAME="E2:12_2_1_1_1_1_17_5"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_17_5"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_17"><i><font color="Green" title="E13">A47</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_17_5"><i><font color="Green" title="E17">A54</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self" title="YELLOW11:th.4">Th4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:12_2_1_1_1_1_17"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E6:12_2_1_1_1_1_17"><i><font color="Green" title="E12">A46</font></i></a>, <a class="txt" href="yellow11.html#E9:12_2_1_1_1_1_17"><i><font color="Green" title="E14">A48</font></i></a>, <a class="txt" href="yellow11.html#E10:12_2_1_1_1_1_17"><i><font color="Green" title="E15">A50</font></i></a>, <a class="txt" href="yellow11.html#E11:12_2_1_1_1_1_17"><i><font color="Green" title="E16">A52</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_18"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_18"/><i><font color="Green" title="E11">A55</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 )
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_1_1_18"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">z</font> = 3 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E4:12_2_1_1_1_1_18"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_18_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_2_1_1_1_1_18_1"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_18"><i><font color="Green" title="E11">A55</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:12_2_1_1_1_1_18_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 ;<br/>

<a NAME="E3:12_2_1_1_1_1_18_1"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_18"><i><font color="Green" title="E11">A55</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:12_2_1_1_1_1_18_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:12_2_1_1_1_1_18"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><a NAME="E6:12_2_1_1_1_1_18"/><i><font color="Green" title="E12">A56</font></i>: 
<font color="Maroon" title="c6">z'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><b>assume </b><a NAME="E7:12_2_1_1_1_1_18"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font> )
 ;<br/><a NAME="E8:12_2_1_1_1_1_18"/><b>then </b><i><font color="Green" title="E13">A57</font></i>: 
( 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z'</font> &amp; 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z'</font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_18"><i><font color="Green" title="E11">A55</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><div><i><font color="Green" title="E14">A58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_18_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_18_2"/><i><font color="Green" title="E15">A59</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:12_2_1_1_1_1_18_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_18_2"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_18"><i><font color="Green" title="E13">A57</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_18_2"><i><font color="Green" title="E15">A59</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E15">A60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_18_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_18_3"/><i><font color="Green" title="E16">A61</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:12_2_1_1_1_1_18_3"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:12_2_1_1_1_1_18_3"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">z'</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_18"><i><font color="Green" title="E13">A57</font></i></a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_18_3"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_18_3"><i><font color="Green" title="E16">A61</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E16">A62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_18_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_18_4"/><i><font color="Green" title="E17">A63</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ;<br/><a NAME="E2:12_2_1_1_1_1_18_4"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 <b>by </b><i><a class="ref" href="yellow11.html#T4" target="_self" title="YELLOW11:th.4">Th4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_18_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_18"><i><font color="Green" title="E13">A57</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_18_4"><i><font color="Green" title="E17">A63</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_18_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1_1_1_1_18_5"/><i><font color="Green" title="E17">A64</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 ;<br/><a NAME="E2:12_2_1_1_1_1_18_5"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_1_18_5"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E8:12_2_1_1_1_1_18"><i><font color="Green" title="E13">A57</font></i></a>, <a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_18_5"><i><font color="Green" title="E17">A64</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self" title="YELLOW11:th.4">Th4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:12_2_1_1_1_1_18"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z'</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E6:12_2_1_1_1_1_18"><i><font color="Green" title="E12">A56</font></i></a>, <a class="txt" href="yellow11.html#E9:12_2_1_1_1_1_18"><i><font color="Green" title="E14">A58</font></i></a>, <a class="txt" href="yellow11.html#E10:12_2_1_1_1_1_18"><i><font color="Green" title="E15">A60</font></i></a>, <a class="txt" href="yellow11.html#E11:12_2_1_1_1_1_18"><i><font color="Green" title="E16">A62</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_19"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_19"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_19"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_19"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_19"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_19"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_19"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_19"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_20"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_20"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_20"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_20"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_20"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_20"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_20"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_20"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_21"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_21"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_21"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_21"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_21"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_21"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_21"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_21"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_22"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_22"/><i><font color="Green" title="E11">A65</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_1_1_22"/>
1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 3
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2_1_1_1_1_22_1"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 ;<br/>

<a NAME="E2:12_2_1_1_1_1_22_1"/><b>then </b>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_1_1_1_1_22_1"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3
 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_22"><i><font color="Green" title="E11">A65</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E4:12_2_1_1_1_1_22"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E5:12_2_1_1_1_1_22"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E6:12_2_1_1_1_1_22"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E7:12_2_1_1_1_1_22"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E8:12_2_1_1_1_1_22"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E9:12_2_1_1_1_1_22"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_23"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_23"/><i><font color="Green" title="E11">A66</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 )
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_1_1_23"/>
2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 3
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2_1_1_1_1_23_1"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ;<br/>

<a NAME="E2:12_2_1_1_1_1_23_1"/><b>then </b>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_1_1_1_1_23_1"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3
 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1_1_1_1_23"><i><font color="Green" title="E11">A66</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E4:12_2_1_1_1_1_23"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E5:12_2_1_1_1_1_23"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E6:12_2_1_1_1_1_23"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E7:12_2_1_1_1_1_23"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E8:12_2_1_1_1_1_23"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E9:12_2_1_1_1_1_23"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_24"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_24"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_24"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_24"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_24"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_24"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_24"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_24"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_25"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_25"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/><a NAME="E3:12_2_1_1_1_1_25"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_1_1_25"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:12_2_1_1_1_1_25"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 ;<br/><a NAME="E6:12_2_1_1_1_1_25"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E7:12_2_1_1_1_1_25"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:12_2_1_1_1_1_25"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E2:12_2_1"/>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:12_2_1_2_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> )
 ;<br/><a NAME="E2:12_2_1_2_1"/><b>then </b><i><font color="Green" title="E10">A67</font></i>: 
( ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp; ( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 <b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><a NAME="E3:12_2_1_2_1"/>
1 <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/><a NAME="E4:12_2_1_2_1"/><b>then </b><i><font color="Green" title="E11">A68</font></i>: 
1 <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:12_2_1_2_1"/><i><font color="Green" title="E12">A69</font></i>: 
1 <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:12_2_1_2_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>
 ;<br/><a NAME="E2:12_2_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:12_2_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 )
 <b>by </b><i><a class="ref" href="yellow11.html#T4" target="_self" title="YELLOW11:th.4">Th4</a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_2_1_2_1_1"/>
1 <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:12_2_1_2_1"/>
1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 3
 <b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/><a NAME="E7:12_2_1_2_1"/><b>then </b><i><font color="Green" title="E13">A70</font></i>: 
1 <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><a NAME="E8:12_2_1_2_1"/><i><font color="Green" title="E14">A71</font></i>: 
<span class="p1">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_2_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:12_2_1_2_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>
 ;<br/><a NAME="E2:12_2_1_2_1_2_1"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:12_2_1_2_1_2_1"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 )
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="yellow11.html#T4" target="_self" title="YELLOW11:th.4">Th4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:12_2_1_2_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_2_1_2_1_2"/>
<span class="p1">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:12_2_1_2_1"/>
2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 3
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2_1_2_1_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ;<br/>

<a NAME="E2:12_2_1_2_1_3"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self" title="YELLOW11:th.2">Th2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_1_2_1_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3
 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:12_2_1_2_1"/><b>then </b><i><font color="Green" title="E15">A72</font></i>: 
<span class="p1">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><a NAME="E11:12_2_1_2_1"/>
<span class="p1">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E12:12_2_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="txt" href="yellow11.html#E2:12_2_1_2_1"><i><font color="Green" title="E10">A67</font></i></a>, <a class="txt" href="yellow11.html#E4:12_2_1_2_1"><i><font color="Green" title="E11">A68</font></i></a>, <a class="txt" href="yellow11.html#E5:12_2_1_2_1"><i><font color="Green" title="E12">A69</font></i></a>, <a class="txt" href="yellow11.html#E7:12_2_1_2_1"><i><font color="Green" title="E13">A70</font></i></a>, <a class="txt" href="yellow11.html#E8:12_2_1_2_1"><i><font color="Green" title="E14">A71</font></i></a>, <a class="txt" href="yellow11.html#E10:12_2_1_2_1"><i><font color="Green" title="E15">A72</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_2_1_2"/>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>
 <b>by </b><i><a class="ref" href="yellow_1.html#T12" title="YELLOW_1:th.12">YELLOW_1:12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:12_2_1"/>
(  <a href="yellow11.html#K2" title="YELLOW11:func.2">M_3</a>  is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a> &amp;  <a href="yellow11.html#K2" title="YELLOW11:func.2">M_3</a>  is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:12_2_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>


</div>
