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

<span class="kw">set </span><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#K5" title="NUMBERS:func.5">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/>
<span class="kw">set </span><font color="Maroon" title="c2">N</font> =  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D10" onmouseover="rs('lattice3/D10')" onmouseout="rh()">LATTICE3:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; (  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>  holds <br/>(  not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/>

<a NAME="E1:12_2_1_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" onmouseover="rs('yellow_1/T1')" onmouseout="rh()">YELLOW_1:1</a></span>;<br/>
<span class="kw">thus </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:12_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#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_1"><span class="kw">suppose </span></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#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_2"><span class="kw">suppose </span></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#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E12">A8</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A9</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E14">A10</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_3"><span class="kw">suppose </span></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#K5" title="NUMBERS:func.5">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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_3"/><span class="lab"><font color="Green" title="E11">A11</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_3"/><span class="lab"><font color="Green" title="E12">A12</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_3"/><span class="lab"><font color="Green" title="E13">A13</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_3"/><span class="lab"><font color="Green" title="E14">A14</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_3"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_4"><span class="kw">suppose </span></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#K5" title="NUMBERS:func.5">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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_4"/><span class="lab"><font color="Green" title="E11">A15</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_4"><span class="lab"><font color="Green" title="E11">A15</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_4"/><span class="lab"><font color="Green" title="E12">A16</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_4"/><span class="lab"><font color="Green" title="E13">A17</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_4"/><span class="lab"><font color="Green" title="E14">A18</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_4"><span class="lab"><font color="Green" title="E13">A17</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_4"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_4"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_4"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_5"><span class="kw">suppose </span></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#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_5"/><span class="lab"><font color="Green" title="E11">A19</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_5"><span class="lab"><font color="Green" title="E11">A19</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_5"/><span class="lab"><font color="Green" title="E12">A20</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_5"/><span class="lab"><font color="Green" title="E13">A21</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_5"/><span class="lab"><font color="Green" title="E14">A22</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_5"><span class="lab"><font color="Green" title="E13">A21</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_5"><span class="lab"><font color="Green" title="E12">A20</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_5"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_5"><span class="lab"><font color="Green" title="E14">A22</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_6"><span class="kw">suppose </span></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#K5" title="NUMBERS:func.5">0</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_6"/><span class="lab"><font color="Green" title="E11">A23</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_6"><span class="lab"><font color="Green" title="E11">A23</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_6"/><span class="lab"><font color="Green" title="E12">A24</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_6"/><span class="lab"><font color="Green" title="E13">A25</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_6"/><span class="lab"><font color="Green" title="E14">A26</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_6"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_6"><span class="lab"><font color="Green" title="E12">A24</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_6"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_6"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_7"><span class="kw">suppose </span></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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_7"/><span class="lab"><font color="Green" title="E11">A27</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_7"><span class="lab"><font color="Green" title="E11">A27</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_7"/><span class="lab"><font color="Green" title="E12">A28</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_7"/><span class="lab"><font color="Green" title="E13">A29</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_7"/><span class="lab"><font color="Green" title="E14">A30</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_7"><span class="lab"><font color="Green" title="E13">A29</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_7"><span class="lab"><font color="Green" title="E12">A28</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_7"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_7"><span class="lab"><font color="Green" title="E14">A30</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_8"><span class="kw">suppose </span></a><a NAME="E1:12_2_1_1_1_1_8"/><span class="lab"><font color="Green" title="E11">A31</font></span>: 
( <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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><a NAME="E2:12_2_1_1_1_1_8"/>
3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" onmouseover="rs('yellow_1/T1')" onmouseout="rh()">YELLOW_1:1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><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>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_8_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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> )</span><br/>

<span class="kw">assume </span><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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font></span><br/>

<a NAME="E2:12_2_1_1_1_1_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E11">A31</font></span></a>, <a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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> )</span><br/>

<span class="kw">assume </span><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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font></span><br/>

<a NAME="E3:12_2_1_1_1_1_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E11">A31</font></span></a>, <a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z9</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">z9</font> implies <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">z9</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_8"/><span class="lab"><font color="Green" title="E12">A32</font></span>: 
<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">z9</font>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_8"/><span class="lab"><font color="Green" title="E13">A33</font></span>: 
<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">z9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">z9</font></span><br/><a NAME="E8:12_2_1_1_1_1_8"/><span class="lab"><font color="Green" title="E14">A34</font></span>: 
<font color="Maroon" title="c6">z9</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#K5" title="NUMBERS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" onmouseover="rs('yellow_1/T1')" onmouseout="rh()">YELLOW_1:1</a></span>;<br/><a NAME="E9:12_2_1_1_1_1_8"/><span class="lab"><font color="Green" title="E15">A35</font></span>: 
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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E11">A31</font></span></a>, <a class="txt" href="#E7:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E13">A33</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E16">A36</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_8_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><a NAME="E1:12_2_1_1_1_1_8_2"/>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E2:12_2_1_1_1_1_8_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A37</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E15">A35</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E3:12_2_1_1_1_1_8_2"/>
<font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:12_2_1_1_1_1_8_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_2_1_1_1_1_8_2"><span class="lab"><font color="Green" title="E17">A37</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:12_2_1_1_1_1_8"/><span class="lab"><font color="Green" title="E17">A38</font></span>: 
1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E11">A31</font></span></a>, <a class="txt" href="#E6:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E12">A32</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E18">A39</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_8_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</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</span></a><div class="add"><a NAME="E1:12_2_1_1_1_1_8_3"/><span class="lab"><font color="Green" title="E19">A40</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:12_2_1_1_1_1_8_3"/>
<font color="Maroon" title="c6">z9</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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_8_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E17">A38</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_8_3"><span class="lab"><font color="Green" title="E19">A40</font></span></a>, <a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E19">A41</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_8_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</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</span></a><div class="add"><a NAME="E1:12_2_1_1_1_1_8_4"/><span class="lab"><font color="Green" title="E20">A42</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:12_2_1_1_1_1_8_4"/>
<font color="Maroon" title="c6">z9</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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_8_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E17">A38</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_8_4"><span class="lab"><font color="Green" title="E20">A42</font></span></a>, <a class="ref" href="yellow11.html#T4" target="_self" onmouseover="rs('yellow11/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E14:12_2_1_1_1_1_8"/>
<font color="Maroon" title="c6">z9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E17">A38</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:12_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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E14">A34</font></span></a>, <a class="txt" href="#E10:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E16">A36</font></span></a>, <a class="txt" href="#E12:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E18">A39</font></span></a>, <a class="txt" href="#E13:12_2_1_1_1_1_8"><span class="lab"><font color="Green" title="E19">A41</font></span></a>, <a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_9"><span class="kw">suppose </span></a><a NAME="E1:12_2_1_1_1_1_9"/><span class="lab"><font color="Green" title="E11">A43</font></span>: 
( <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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><a NAME="E2:12_2_1_1_1_1_9"/>
3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" onmouseover="rs('yellow_1/T1')" onmouseout="rh()">YELLOW_1:1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><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>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_2_1_1_1_1_9_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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font></span><br/>

<a NAME="E2:12_2_1_1_1_1_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E11">A43</font></span></a>, <a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_9_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E11">A43</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z9</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">z9</font> implies <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">z9</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_9"/><span class="lab"><font color="Green" title="E12">A44</font></span>: 
<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">z9</font>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_9"/><span class="lab"><font color="Green" title="E13">A45</font></span>: 
<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">z9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">z9</font></span><br/><a NAME="E8:12_2_1_1_1_1_9"/><span class="lab"><font color="Green" title="E14">A46</font></span>: 
<font color="Maroon" title="c6">z9</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#K5" title="NUMBERS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" onmouseover="rs('yellow_1/T1')" onmouseout="rh()">YELLOW_1:1</a></span>;<br/><a NAME="E9:12_2_1_1_1_1_9"/><span class="lab"><font color="Green" title="E15">A47</font></span>: 
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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E11">A43</font></span></a>, <a class="txt" href="#E7:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E13">A45</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E16">A48</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_9_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:12_2_1_1_1_1_9_2"/><span class="lab"><font color="Green" title="E17">A49</font></span>: 
<font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:12_2_1_1_1_1_9_2"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T4" target="_self" onmouseover="rs('yellow11/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_9_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E15">A47</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_9_2"><span class="lab"><font color="Green" title="E17">A49</font></span></a>, <a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:12_2_1_1_1_1_9"/><span class="lab"><font color="Green" title="E17">A50</font></span>: 
1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E11">A43</font></span></a>, <a class="txt" href="#E6:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E12">A44</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E18">A51</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_9_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</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</span></a><div class="add"><a NAME="E1:12_2_1_1_1_1_9_3"/><span class="lab"><font color="Green" title="E19">A52</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:12_2_1_1_1_1_9_3"/>
<font color="Maroon" title="c6">z9</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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_9_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E17">A50</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_9_3"><span class="lab"><font color="Green" title="E19">A52</font></span></a>, <a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E19">A53</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_9_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</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</span></a><div class="add"><a NAME="E1:12_2_1_1_1_1_9_4"/><span class="lab"><font color="Green" title="E20">A54</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:12_2_1_1_1_1_9_4"/>
<font color="Maroon" title="c6">z9</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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_9_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E17">A50</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_9_4"><span class="lab"><font color="Green" title="E20">A54</font></span></a>, <a class="ref" href="yellow11.html#T4" target="_self" onmouseover="rs('yellow11/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E14:12_2_1_1_1_1_9"/>
<font color="Maroon" title="c6">z9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E17">A50</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:12_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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E14">A46</font></span></a>, <a class="txt" href="#E10:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E16">A48</font></span></a>, <a class="txt" href="#E12:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E18">A51</font></span></a>, <a class="txt" href="#E13:12_2_1_1_1_1_9"><span class="lab"><font color="Green" title="E19">A53</font></span></a>, <a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_10"><span class="kw">suppose </span></a><a NAME="E1:12_2_1_1_1_1_10"/><span class="lab"><font color="Green" title="E11">A55</font></span>: 
( <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 class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><a NAME="E2:12_2_1_1_1_1_10"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 3
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 1 or <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 3 )</span><br/>

<span class="kw">assume </span><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>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 3</span><br/>

<a NAME="E2:12_2_1_1_1_1_10_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12_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>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_10"><span class="lab"><font color="Green" title="E11">A55</font></span></a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E4:12_2_1_1_1_1_10"/><span class="lab"><font color="Green" title="E12">A56</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E5:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_2_1_1_1_1_10"><span class="lab"><font color="Green" title="E12">A56</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:12_2_1_1_1_1_10"/><span class="lab"><font color="Green" title="E13">A57</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:12_2_1_1_1_1_10"/><span class="lab"><font color="Green" title="E14">A58</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E9:12_2_1_1_1_1_10"/><span class="lab"><font color="Green" title="E15">A59</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_10"><span class="lab"><font color="Green" title="E14">A58</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_10"><span class="lab"><font color="Green" title="E13">A57</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E11:12_2_1_1_1_1_10"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_2_1_1_1_1_10"><span class="lab"><font color="Green" title="E15">A59</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E12:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_11"><span class="kw">suppose </span></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#K5" title="NUMBERS:func.5">0</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_11"/><span class="lab"><font color="Green" title="E11">A60</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_11"><span class="lab"><font color="Green" title="E11">A60</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_11"/><span class="lab"><font color="Green" title="E12">A61</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_11"/><span class="lab"><font color="Green" title="E13">A62</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_11"/><span class="lab"><font color="Green" title="E14">A63</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_11"><span class="lab"><font color="Green" title="E13">A62</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_11"><span class="lab"><font color="Green" title="E12">A61</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_11"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_11"><span class="lab"><font color="Green" title="E14">A63</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_12"><span class="kw">suppose </span></a><a NAME="E1:12_2_1_1_1_1_12"/><span class="lab"><font color="Green" title="E11">A64</font></span>: 
( <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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><a NAME="E2:12_2_1_1_1_1_12"/>
3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" onmouseover="rs('yellow_1/T1')" onmouseout="rh()">YELLOW_1:1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_12_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><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>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_12_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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> )</span><br/>

<span class="kw">assume </span><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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font></span><br/>

<a NAME="E2:12_2_1_1_1_1_12_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E11">A64</font></span></a>, <a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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> )</span><br/>

<span class="kw">assume </span><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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font></span><br/>

<a NAME="E3:12_2_1_1_1_1_12_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E11">A64</font></span></a>, <a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z9</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">z9</font> implies <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">z9</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_12"/><span class="lab"><font color="Green" title="E12">A65</font></span>: 
<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">z9</font>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_12"/><span class="lab"><font color="Green" title="E13">A66</font></span>: 
<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">z9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">z9</font></span><br/><a NAME="E8:12_2_1_1_1_1_12"/><span class="lab"><font color="Green" title="E14">A67</font></span>: 
<font color="Maroon" title="c6">z9</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#K5" title="NUMBERS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" onmouseover="rs('yellow_1/T1')" onmouseout="rh()">YELLOW_1:1</a></span>;<br/><a NAME="E9:12_2_1_1_1_1_12"/><span class="lab"><font color="Green" title="E15">A68</font></span>: 
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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E11">A64</font></span></a>, <a class="txt" href="#E6:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E12">A65</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E16">A69</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_12_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><a NAME="E1:12_2_1_1_1_1_12_2"/>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E2:12_2_1_1_1_1_12_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A70</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E15">A68</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E3:12_2_1_1_1_1_12_2"/>
<font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:12_2_1_1_1_1_12_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_2_1_1_1_1_12_2"><span class="lab"><font color="Green" title="E17">A70</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:12_2_1_1_1_1_12"/><span class="lab"><font color="Green" title="E17">A71</font></span>: 
1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E11">A64</font></span></a>, <a class="txt" href="#E7:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E13">A66</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E18">A72</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_12_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</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</span></a><div class="add"><a NAME="E1:12_2_1_1_1_1_12_3"/><span class="lab"><font color="Green" title="E19">A73</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:12_2_1_1_1_1_12_3"/>
<font color="Maroon" title="c6">z9</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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_12_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E17">A71</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_12_3"><span class="lab"><font color="Green" title="E19">A73</font></span></a>, <a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E19">A74</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_12_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</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</span></a><div class="add"><a NAME="E1:12_2_1_1_1_1_12_4"/><span class="lab"><font color="Green" title="E20">A75</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:12_2_1_1_1_1_12_4"/>
<font color="Maroon" title="c6">z9</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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_12_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E17">A71</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_12_4"><span class="lab"><font color="Green" title="E20">A75</font></span></a>, <a class="ref" href="yellow11.html#T4" target="_self" onmouseover="rs('yellow11/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E14:12_2_1_1_1_1_12"/>
<font color="Maroon" title="c6">z9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E17">A71</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:12_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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E14">A67</font></span></a>, <a class="txt" href="#E10:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E16">A69</font></span></a>, <a class="txt" href="#E12:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E18">A72</font></span></a>, <a class="txt" href="#E13:12_2_1_1_1_1_12"><span class="lab"><font color="Green" title="E19">A74</font></span></a>, <a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_13"><span class="kw">suppose </span></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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_13"/><span class="lab"><font color="Green" title="E11">A76</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_13"><span class="lab"><font color="Green" title="E11">A76</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_13"/><span class="lab"><font color="Green" title="E12">A77</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_13"/><span class="lab"><font color="Green" title="E13">A78</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_13"/><span class="lab"><font color="Green" title="E14">A79</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_13"><span class="lab"><font color="Green" title="E13">A78</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_13"><span class="lab"><font color="Green" title="E12">A77</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_13"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_13"><span class="lab"><font color="Green" title="E14">A79</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_14"><span class="kw">suppose </span></a><a NAME="E1:12_2_1_1_1_1_14"/><span class="lab"><font color="Green" title="E11">A80</font></span>: 
( <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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><a NAME="E2:12_2_1_1_1_1_14"/>
3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" onmouseover="rs('yellow_1/T1')" onmouseout="rh()">YELLOW_1:1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><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>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_14_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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> )</span><br/>

<span class="kw">assume </span><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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font></span><br/>

<a NAME="E2:12_2_1_1_1_1_14_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E11">A80</font></span></a>, <a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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> )</span><br/>

<span class="kw">assume </span><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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font></span><br/>

<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E11">A80</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z9</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">z9</font> implies <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">z9</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_14"/><span class="lab"><font color="Green" title="E12">A81</font></span>: 
<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">z9</font>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_14"/><span class="lab"><font color="Green" title="E13">A82</font></span>: 
<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">z9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">z9</font></span><br/><a NAME="E8:12_2_1_1_1_1_14"/><span class="lab"><font color="Green" title="E14">A83</font></span>: 
<font color="Maroon" title="c6">z9</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#K5" title="NUMBERS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" onmouseover="rs('yellow_1/T1')" onmouseout="rh()">YELLOW_1:1</a></span>;<br/><a NAME="E9:12_2_1_1_1_1_14"/><span class="lab"><font color="Green" title="E15">A84</font></span>: 
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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E11">A80</font></span></a>, <a class="txt" href="#E7:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E13">A82</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E16">A85</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_14_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</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</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:12_2_1_1_1_1_14_2"/><span class="lab"><font color="Green" title="E17">A86</font></span>: 
<font color="Maroon" title="c6">z9</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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:12_2_1_1_1_1_14_2"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T4" target="_self" onmouseover="rs('yellow11/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_14_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E15">A84</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_14_2"><span class="lab"><font color="Green" title="E17">A86</font></span></a>, <a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:12_2_1_1_1_1_14"/><span class="lab"><font color="Green" title="E17">A87</font></span>: 
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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E11">A80</font></span></a>, <a class="txt" href="#E6:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E12">A81</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E18">A88</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_14_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</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</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:12_2_1_1_1_1_14_3"/><span class="lab"><font color="Green" title="E19">A89</font></span>: 
<font color="Maroon" title="c6">z9</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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:12_2_1_1_1_1_14_3"/>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_14_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E17">A87</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_14_3"><span class="lab"><font color="Green" title="E19">A89</font></span></a>, <a class="ref" href="yellow11.html#T4" target="_self" onmouseover="rs('yellow11/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E19">A90</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_14_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><a NAME="E1:12_2_1_1_1_1_14_4"/>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E2:12_2_1_1_1_1_14_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A91</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E17">A87</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E3:12_2_1_1_1_1_14_4"/>
<font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:12_2_1_1_1_1_14_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_2_1_1_1_1_14_4"><span class="lab"><font color="Green" title="E20">A91</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E14:12_2_1_1_1_1_14"/>
<font color="Maroon" title="c6">z9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E17">A87</font></span></a>, <a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a></span>;<br/><span class="kw">hence </span><a NAME="E15:12_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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E14">A83</font></span></a>, <a class="txt" href="#E13:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E19">A90</font></span></a>, <a class="txt" href="#E10:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E16">A85</font></span></a>, <a class="txt" href="#E12:12_2_1_1_1_1_14"><span class="lab"><font color="Green" title="E18">A88</font></span></a>, <a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_15"><span class="kw">suppose </span></a><a NAME="E1:12_2_1_1_1_1_15"/><span class="lab"><font color="Green" title="E11">A92</font></span>: 
( <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 class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_15_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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 or <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 )</span><br/>

<span class="kw">assume </span><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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3</span><br/>

<a NAME="E2:12_2_1_1_1_1_15_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12_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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_15"><span class="lab"><font color="Green" title="E11">A92</font></span></a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E4:12_2_1_1_1_1_15"/><span class="lab"><font color="Green" title="E12">A93</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E5:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_2_1_1_1_1_15"><span class="lab"><font color="Green" title="E12">A93</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:12_2_1_1_1_1_15"/><span class="lab"><font color="Green" title="E13">A94</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:12_2_1_1_1_1_15"/><span class="lab"><font color="Green" title="E14">A95</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E9:12_2_1_1_1_1_15"/><span class="lab"><font color="Green" title="E15">A96</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_15"><span class="lab"><font color="Green" title="E14">A95</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_15"><span class="lab"><font color="Green" title="E13">A94</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E11:12_2_1_1_1_1_15"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_2_1_1_1_1_15"><span class="lab"><font color="Green" title="E15">A96</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E12:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_16"><span class="kw">suppose </span></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#K5" title="NUMBERS:func.5">0</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_16"/><span class="lab"><font color="Green" title="E11">A97</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_16"><span class="lab"><font color="Green" title="E11">A97</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_16"/><span class="lab"><font color="Green" title="E12">A98</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_16"/><span class="lab"><font color="Green" title="E13">A99</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_16"/><span class="lab"><font color="Green" title="E14">A100</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_16"><span class="lab"><font color="Green" title="E13">A99</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_16"><span class="lab"><font color="Green" title="E12">A98</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_16"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_16"><span class="lab"><font color="Green" title="E14">A100</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_17"><span class="kw">suppose </span></a><a NAME="E1:12_2_1_1_1_1_17"/><span class="lab"><font color="Green" title="E11">A101</font></span>: 
( <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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><a NAME="E2:12_2_1_1_1_1_17"/>
3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" onmouseover="rs('yellow_1/T1')" onmouseout="rh()">YELLOW_1:1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E4:12_2_1_1_1_1_17"/>
<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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_2_1_1_1_1_17_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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font></span><br/>

<a NAME="E2:12_2_1_1_1_1_17_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E11">A101</font></span></a>, <a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_17_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E11">A101</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z9</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">z9</font> implies <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">z9</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_17"/><span class="lab"><font color="Green" title="E12">A102</font></span>: 
<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">z9</font>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_17"/><span class="lab"><font color="Green" title="E13">A103</font></span>: 
<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">z9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">z9</font></span><br/><a NAME="E8:12_2_1_1_1_1_17"/><span class="lab"><font color="Green" title="E14">A104</font></span>: 
<font color="Maroon" title="c6">z9</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#K5" title="NUMBERS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" onmouseover="rs('yellow_1/T1')" onmouseout="rh()">YELLOW_1:1</a></span>;<br/><a NAME="E9:12_2_1_1_1_1_17"/><span class="lab"><font color="Green" title="E15">A105</font></span>: 
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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E11">A101</font></span></a>, <a class="txt" href="#E6:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E12">A102</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E16">A106</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_17_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:12_2_1_1_1_1_17_2"/><span class="lab"><font color="Green" title="E17">A107</font></span>: 
<font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:12_2_1_1_1_1_17_2"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T4" target="_self" onmouseover="rs('yellow11/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_17_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E15">A105</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_17_2"><span class="lab"><font color="Green" title="E17">A107</font></span></a>, <a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:12_2_1_1_1_1_17"/><span class="lab"><font color="Green" title="E17">A108</font></span>: 
1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E11">A101</font></span></a>, <a class="txt" href="#E7:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E13">A103</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E18">A109</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_17_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</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</span></a><div class="add"><a NAME="E1:12_2_1_1_1_1_17_3"/><span class="lab"><font color="Green" title="E19">A110</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:12_2_1_1_1_1_17_3"/>
<font color="Maroon" title="c6">z9</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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_17_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E17">A108</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_17_3"><span class="lab"><font color="Green" title="E19">A110</font></span></a>, <a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E19">A111</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_17_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</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</span></a><div class="add"><a NAME="E1:12_2_1_1_1_1_17_4"/><span class="lab"><font color="Green" title="E20">A112</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:12_2_1_1_1_1_17_4"/>
<font color="Maroon" title="c6">z9</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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_17_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E17">A108</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_17_4"><span class="lab"><font color="Green" title="E20">A112</font></span></a>, <a class="ref" href="yellow11.html#T4" target="_self" onmouseover="rs('yellow11/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E14:12_2_1_1_1_1_17"/>
<font color="Maroon" title="c6">z9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E17">A108</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:12_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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E14">A104</font></span></a>, <a class="txt" href="#E10:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E16">A106</font></span></a>, <a class="txt" href="#E12:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E18">A109</font></span></a>, <a class="txt" href="#E13:12_2_1_1_1_1_17"><span class="lab"><font color="Green" title="E19">A111</font></span></a>, <a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_18"><span class="kw">suppose </span></a><a NAME="E1:12_2_1_1_1_1_18"/><span class="lab"><font color="Green" title="E11">A113</font></span>: 
( <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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><a NAME="E2:12_2_1_1_1_1_18"/>
3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" onmouseover="rs('yellow_1/T1')" onmouseout="rh()">YELLOW_1:1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E11">A113</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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> )</span><br/>

<span class="kw">assume </span><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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font></span><br/>

<a NAME="E3:12_2_1_1_1_1_18_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E11">A113</font></span></a>, <a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">z9</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">z9</font> implies <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">z9</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_18"/><span class="lab"><font color="Green" title="E12">A114</font></span>: 
<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">z9</font>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_18"/><span class="lab"><font color="Green" title="E13">A115</font></span>: 
<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">z9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">z9</font></span><br/><a NAME="E8:12_2_1_1_1_1_18"/><span class="lab"><font color="Green" title="E14">A116</font></span>: 
<font color="Maroon" title="c6">z9</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#K5" title="NUMBERS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" onmouseover="rs('yellow_1/T1')" onmouseout="rh()">YELLOW_1:1</a></span>;<br/><a NAME="E9:12_2_1_1_1_1_18"/><span class="lab"><font color="Green" title="E15">A117</font></span>: 
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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E11">A113</font></span></a>, <a class="txt" href="#E6:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E12">A114</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E16">A118</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_18_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</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</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:12_2_1_1_1_1_18_2"/><span class="lab"><font color="Green" title="E17">A119</font></span>: 
<font color="Maroon" title="c6">z9</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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:12_2_1_1_1_1_18_2"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T4" target="_self" onmouseover="rs('yellow11/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_18_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E15">A117</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_18_2"><span class="lab"><font color="Green" title="E17">A119</font></span></a>, <a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:12_2_1_1_1_1_18"/><span class="lab"><font color="Green" title="E17">A120</font></span>: 
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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E11">A113</font></span></a>, <a class="txt" href="#E7:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E13">A115</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E18">A121</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_18_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</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</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:12_2_1_1_1_1_18_3"/><span class="lab"><font color="Green" title="E19">A122</font></span>: 
<font color="Maroon" title="c6">z9</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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:12_2_1_1_1_1_18_3"/>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1_1_18_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E17">A120</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1_1_18_3"><span class="lab"><font color="Green" title="E19">A122</font></span></a>, <a class="ref" href="yellow11.html#T4" target="_self" onmouseover="rs('yellow11/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E19">A123</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_18_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><a NAME="E1:12_2_1_1_1_1_18_4"/>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E2:12_2_1_1_1_1_18_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A124</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E17">A120</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E3:12_2_1_1_1_1_18_4"/>
<font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:12_2_1_1_1_1_18_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_2_1_1_1_1_18_4"><span class="lab"><font color="Green" title="E20">A124</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E14:12_2_1_1_1_1_18"/>
<font color="Maroon" title="c6">z9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E17">A120</font></span></a>, <a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a></span>;<br/><span class="kw">hence </span><a NAME="E15:12_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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E14">A116</font></span></a>, <a class="txt" href="#E13:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E19">A123</font></span></a>, <a class="txt" href="#E10:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E16">A118</font></span></a>, <a class="txt" href="#E12:12_2_1_1_1_1_18"><span class="lab"><font color="Green" title="E18">A121</font></span></a>, <a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_19"><span class="kw">suppose </span></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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_19"/><span class="lab"><font color="Green" title="E11">A125</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_19"><span class="lab"><font color="Green" title="E11">A125</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_19"/><span class="lab"><font color="Green" title="E12">A126</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_19"/><span class="lab"><font color="Green" title="E13">A127</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_19"/><span class="lab"><font color="Green" title="E14">A128</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_19"><span class="lab"><font color="Green" title="E13">A127</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_19"><span class="lab"><font color="Green" title="E12">A126</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_19"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_19"><span class="lab"><font color="Green" title="E14">A128</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_20"><span class="kw">suppose </span></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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_20"/><span class="lab"><font color="Green" title="E11">A129</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_20"><span class="lab"><font color="Green" title="E11">A129</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_20"/><span class="lab"><font color="Green" title="E12">A130</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_20"/><span class="lab"><font color="Green" title="E13">A131</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_20"/><span class="lab"><font color="Green" title="E14">A132</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_20"><span class="lab"><font color="Green" title="E13">A131</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_20"><span class="lab"><font color="Green" title="E12">A130</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_20"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_20"><span class="lab"><font color="Green" title="E14">A132</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_21"><span class="kw">suppose </span></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#K5" title="NUMBERS:func.5">0</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_21"/><span class="lab"><font color="Green" title="E11">A133</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_21"><span class="lab"><font color="Green" title="E11">A133</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_21"/><span class="lab"><font color="Green" title="E12">A134</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_21"/><span class="lab"><font color="Green" title="E13">A135</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_21"/><span class="lab"><font color="Green" title="E14">A136</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_21"><span class="lab"><font color="Green" title="E13">A135</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_21"><span class="lab"><font color="Green" title="E12">A134</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_21"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_21"><span class="lab"><font color="Green" title="E14">A136</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_22"><span class="kw">suppose </span></a><a NAME="E1:12_2_1_1_1_1_22"/><span class="lab"><font color="Green" title="E11">A137</font></span>: 
( <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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><a NAME="E2:12_2_1_1_1_1_22"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 3
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_22_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 1 or <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 3 )</span><br/>

<span class="kw">assume </span><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>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 3</span><br/>

<a NAME="E2:12_2_1_1_1_1_22_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12_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>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_22"><span class="lab"><font color="Green" title="E11">A137</font></span></a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E4:12_2_1_1_1_1_22"/><span class="lab"><font color="Green" title="E12">A138</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E5:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_2_1_1_1_1_22"><span class="lab"><font color="Green" title="E12">A138</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:12_2_1_1_1_1_22"/><span class="lab"><font color="Green" title="E13">A139</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:12_2_1_1_1_1_22"/><span class="lab"><font color="Green" title="E14">A140</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E9:12_2_1_1_1_1_22"/><span class="lab"><font color="Green" title="E15">A141</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_22"><span class="lab"><font color="Green" title="E14">A140</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_22"><span class="lab"><font color="Green" title="E13">A139</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E11:12_2_1_1_1_1_22"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_2_1_1_1_1_22"><span class="lab"><font color="Green" title="E15">A141</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E12:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_23"><span class="kw">suppose </span></a><a NAME="E1:12_2_1_1_1_1_23"/><span class="lab"><font color="Green" title="E11">A142</font></span>: 
( <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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_23_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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 or <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 )</span><br/>

<span class="kw">assume </span><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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3</span><br/>

<a NAME="E2:12_2_1_1_1_1_23_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12_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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_1_1_1_23"><span class="lab"><font color="Green" title="E11">A142</font></span></a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E4:12_2_1_1_1_1_23"/><span class="lab"><font color="Green" title="E12">A143</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E5:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_2_1_1_1_1_23"><span class="lab"><font color="Green" title="E12">A143</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:12_2_1_1_1_1_23"/><span class="lab"><font color="Green" title="E13">A144</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:12_2_1_1_1_1_23"/><span class="lab"><font color="Green" title="E14">A145</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E9:12_2_1_1_1_1_23"/><span class="lab"><font color="Green" title="E15">A146</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_23"><span class="lab"><font color="Green" title="E14">A145</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_23"><span class="lab"><font color="Green" title="E13">A144</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E11:12_2_1_1_1_1_23"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_2_1_1_1_1_23"><span class="lab"><font color="Green" title="E15">A146</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E12:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_24"><span class="kw">suppose </span></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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_24"/><span class="lab"><font color="Green" title="E11">A147</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_24"><span class="lab"><font color="Green" title="E11">A147</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_24"/><span class="lab"><font color="Green" title="E12">A148</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_24"/><span class="lab"><font color="Green" title="E13">A149</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_24"/><span class="lab"><font color="Green" title="E14">A150</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_24"><span class="lab"><font color="Green" title="E13">A149</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_24"><span class="lab"><font color="Green" title="E12">A148</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_24"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_24"><span class="lab"><font color="Green" title="E14">A150</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_25"><span class="kw">suppose </span></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 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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">z9</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">z9</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">z9</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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/><span class="kw">take </span>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font> ) )</span><br/><a NAME="E3:12_2_1_1_1_1_25"/><span class="lab"><font color="Green" title="E11">A151</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E4:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_1_1_1_25"><span class="lab"><font color="Green" title="E11">A151</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z9</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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="b1">z9</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">z9</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z9</font></span><br/><span class="kw">let </span><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#K1" title="YELLOW_1:func.1">InclPoset</a> <span class="p2"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:12_2_1_1_1_1_25"/><span class="lab"><font color="Green" title="E12">A152</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:12_2_1_1_1_1_25"/><span class="lab"><font color="Green" title="E13">A153</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/><a NAME="E8:12_2_1_1_1_1_25"/><span class="lab"><font color="Green" title="E14">A154</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_2_1_1_1_1_25"><span class="lab"><font color="Green" title="E13">A153</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E9:12_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="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2_1_1_1_1_25"><span class="lab"><font color="Green" title="E12">A152</font></span></a>, <a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>;<br/><a NAME="E10:12_2_1_1_1_1_25"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2_1_1_1_1_25"><span class="lab"><font color="Green" title="E14">A154</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:12_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">YELLOW_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> holds <br/><font color="Olive" title="b1">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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></span></a><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">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></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">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></span><br/><span class="kw">assume </span><a NAME="E1:12_2_1_2_1"/><span class="lab"><font color="Green" title="E10">A155</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">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 class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:12_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:12_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A156</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E4:12_2_1_2_1"/>
<font color="Maroon" title="c3">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_2_1"><span class="lab"><font color="Green" title="E10">A155</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:12_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_2_1"><span class="lab"><font color="Green" title="E11">A156</font></span></a>, <a class="ref" href="yellow11.html#T4" target="_self" onmouseover="rs('yellow11/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E2:12_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A157</font></span>: 
<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#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><a NAME="E3:12_2_1_2"/><span class="lab"><font color="Green" title="E11">A158</font></span>: 
<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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/><a NAME="E4:12_2_1_2"/>
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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">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 or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_2_1_2_2"/>
<font color="Maroon" title="c3">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
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3</span><br/>

<a NAME="E2:12_2_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T2" target="_self" onmouseover="rs('yellow11/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12_2_1_2_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:12_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A159</font></span>: 
<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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/><a NAME="E6:12_2_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" onmouseover="rs('nat_1/T39')" onmouseout="rh()">NAT_1:39</a></span>;<br/><a NAME="E7:12_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A160</font></span>: 
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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1_2_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">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></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">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></span><br/><span class="kw">assume </span><a NAME="E1:12_2_1_2_3"/><span class="lab"><font color="Green" title="E14">A161</font></span>: 
<font color="Maroon" title="c3">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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:12_2_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:12_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A162</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E4:12_2_1_2_3"/>
<font color="Maroon" title="c3">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1_2_3"><span class="lab"><font color="Green" title="E14">A161</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:12_2_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2_1_2_3"><span class="lab"><font color="Green" title="E15">A162</font></span></a>, <a class="ref" href="yellow11.html#T4" target="_self" onmouseover="rs('yellow11/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:12_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A163</font></span>: 
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#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><a NAME="E10:12_2_1_2"/>
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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a></span>;<br/><a NAME="E11:12_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A164</font></span>: 
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#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> implies <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="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E12:12_2_1_2"/><span class="lab"><font color="Green" title="E16">A165</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>
 <span class="kw">and </span><br/><a NAME="E13:12_2_1_2"/><span class="lab"><font color="Green" title="E17">A166</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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 class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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></span><br/><a NAME="E14:12_2_1_2"/><span class="lab"><font color="Green" title="E18">A167</font></span>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">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 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:12_2_1_2"><span class="lab"><font color="Green" title="E17">A166</font></span></a>, <a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>;<br/><a NAME="E15:12_2_1_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">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 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:12_2_1_2"><span class="lab"><font color="Green" title="E16">A165</font></span></a>, <a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E16:12_2_1_2"/>
<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="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:12_2_1_2"><span class="lab"><font color="Green" title="E18">A167</font></span></a>, <a class="txt" href="#E11:12_2_1_2"><span class="lab"><font color="Green" title="E15">A164</font></span></a>, <a class="txt" href="#E9:12_2_1_2"><span class="lab"><font color="Green" title="E14">A163</font></span></a>, <a class="txt" href="#E7:12_2_1_2"><span class="lab"><font color="Green" title="E13">A160</font></span></a>, <a class="txt" href="#E2:12_2_1_2"><span class="lab"><font color="Green" title="E10">A157</font></span></a>, <a class="txt" href="#E5:12_2_1_2"><span class="lab"><font color="Green" title="E12">A159</font></span></a>, <a class="txt" href="#E3:12_2_1_2"><span class="lab"><font color="Green" title="E11">A158</font></span></a>, <a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:12_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="yellow_1.html#T12" onmouseover="rs('yellow_1/T12')" onmouseout="rh()">YELLOW_1:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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