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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</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="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies ( <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) )</span><br/>

<a NAME="E1:30"/><i><font color="Green" title="E23">Z1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,1,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,2,<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#K4" title="ENUMSET1:func.4">}</a></span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<b>assume </b><a NAME="E2:30"/><i><font color="Green" title="E24">Z</font></i>: 
( <font color="Maroon" title="c1">a</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="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )</span><br/>

<a NAME="E3:30"/>
( 3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,1,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,2,<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#K4" title="ENUMSET1:func.4">}</a></span> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,1,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1</span>)</span>,2,<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#K4" title="ENUMSET1:func.4">}</a></span> )
 
<b>by </b><i><a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">t</font> = 3, <font color="Maroon" title="c4">z</font> =  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a>  <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E5:30"/>
( <font color="Maroon" title="c1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">t</font> )
 
<b>by </b><i><a class="txt" href="#E2:30"><i><font color="Green" title="E24">Z</font></i></a>, <a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<a NAME="E6:30"/><b>then </b><i><font color="Green" title="E25">A0</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c2">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">t</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<a NAME="E7:30"/><i><font color="Green" title="E26">a1</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a>   st <font color="Olive" title="b1">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Olive" title="b1">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font> holds <br/><font color="Maroon" title="c3">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">z'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c5">z'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font> implies <font color="Maroon" title="c3">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z'</font> )</span><br/>

<b>assume </b><a NAME="E1:30_1"/>
( <font color="Maroon" title="c5">z'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c5">z'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z'</font></span><br/>

<a NAME="E2:30_1"/><b>then </b><i><font color="Green" title="E27">A11</font></i>: 
( 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="c5">z'</font> &amp; 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">z'</font> )
 
<b>by </b><i><a class="txt" href="#E2:30"><i><font color="Green" title="E24">Z</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<div><i><font color="Green" title="E28">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c5">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:30_1_1"/><b>then </b>
( not  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z'</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 )
 <b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="yellow11.html#T4" title="YELLOW11:th.4">YELLOW11:4</a></i>;<br/><b>hence </b><a NAME="E3:30_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:30_1"><i><font color="Green" title="E27">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E29">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_2"/>
<font color="Maroon" title="c5">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:30_1_2"/><b>then </b>
( not 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z'</font> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="yellow11.html#T4" title="YELLOW11:th.4">YELLOW11:4</a></i>;<br/><b>hence </b><a NAME="E3:30_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:30_1"><i><font color="Green" title="E27">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E30">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_3"/>
<font color="Maroon" title="c5">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:30_1_3"/><b>then </b>
( not  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z'</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 )
 <b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="yellow11.html#T3" title="YELLOW11:th.3">YELLOW11:3</a></i>;<br/><b>hence </b><a NAME="E3:30_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:30_1"><i><font color="Green" title="E27">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E31">A00</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_4"/>
<font color="Maroon" title="c5">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:30_1_4"/><b>then </b>
( 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 &amp; not 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z'</font> )
 <b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="yellow11.html#T4" title="YELLOW11:th.4">YELLOW11:4</a></i>;<br/><b>hence </b><a NAME="E3:30_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:30_1"><i><font color="Green" title="E27">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E7:30_1"/>
<font color="Maroon" title="c5">z'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="#E2:30_1"><i><font color="Green" title="E27">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:30_1"/>
<font color="Maroon" title="c3">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z'</font>
 <b>by </b><i><a class="txt" href="#E6:30_1"><i><font color="Green" title="E31">A00</font></i></a>, <a class="txt" href="#E3:30_1"><i><font color="Green" title="E28">A12</font></i></a>, <a class="txt" href="#E4:30_1"><i><font color="Green" title="E29">A13</font></i></a>, <a class="txt" href="#E5:30_1"><i><font color="Green" title="E30">A14</font></i></a>, <a class="txt" href="#E1:30"><i><font color="Green" title="E23">Z1</font></i></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:30"/>
( <font color="Maroon" title="c4">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c4">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E9:30"/><b>then </b><i><font color="Green" title="E27">Z0</font></i>: 
( <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<a NAME="E10:30"/>
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a>   st <font color="Maroon" title="c1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">d</font> holds <br/><font color="Olive" title="b1">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">z'</font> &amp; <font color="Maroon" title="c2">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">z'</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="c4">z</font> )</span><br/>

<b>assume </b><a NAME="E1:30_2"/>
( <font color="Maroon" title="c1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">z'</font> &amp; <font color="Maroon" title="c2">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">z'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c4">z</font></span><br/>

<a NAME="E2:30_2"/><b>then </b><i><font color="Green" title="E28">A11</font></i>: 
( <font color="Maroon" title="c5">z'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 &amp; <font color="Maroon" title="c5">z'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 1 )
 
<b>by </b><i><a class="txt" href="#E2:30"><i><font color="Green" title="E24">Z</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<div><i><font color="Green" title="E29">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2_1"/>
<font color="Maroon" title="c5">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:30_2_1"/><b>then </b>
( 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z'</font> &amp; not 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 )
 <b>by </b><i><a class="ref" href="card_1.html#T88" title="CARD_1:th.88">CARD_1:88</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="yellow11.html#T3" title="YELLOW11:th.3">YELLOW11:3</a></i>;<br/><b>hence </b><a NAME="E3:30_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:30_2"><i><font color="Green" title="E28">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E30">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2_2"/>
<font color="Maroon" title="c5">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:30_2_2"/><b>then </b>
(  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z'</font> &amp; not  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 )
 <b>by </b><i><a class="ref" href="card_1.html#T88" title="CARD_1:th.88">CARD_1:88</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="yellow11.html#T4" title="YELLOW11:th.4">YELLOW11:4</a></i>;<br/><b>hence </b><a NAME="E3:30_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:30_2"><i><font color="Green" title="E28">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E31">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2_3"/>
<font color="Maroon" title="c5">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:30_2_3"/><b>then </b>
( 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z'</font> &amp; not 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 )
 <b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="yellow11.html#T4" title="YELLOW11:th.4">YELLOW11:4</a></i>;<br/><b>hence </b><a NAME="E3:30_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:30_2"><i><font color="Green" title="E28">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E32">A00</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2_4"/>
<font color="Maroon" title="c5">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:30_2_4"/><b>then </b>
(  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z'</font> &amp; not  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 )
 <b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="yellow11.html#T4" title="YELLOW11:th.4">YELLOW11:4</a></i>;<br/><b>hence </b><a NAME="E3:30_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:30_2"><i><font color="Green" title="E28">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2_5"/>
<font color="Maroon" title="c5">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:30_2_5"/><b>then </b>
( 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z'</font> &amp; not 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 )
 <b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:30_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:30_2"><i><font color="Green" title="E28">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:30_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="c4">z</font>
 <b>by </b><i><a class="txt" href="#E6:30_2"><i><font color="Green" title="E32">A00</font></i></a>, <a class="txt" href="#E3:30_2"><i><font color="Green" title="E29">A12</font></i></a>, <a class="txt" href="#E4:30_2"><i><font color="Green" title="E30">A13</font></i></a>, <a class="txt" href="#E5:30_2"><i><font color="Green" title="E31">A14</font></i></a>, <a class="txt" href="#E1:30"><i><font color="Green" title="E23">Z1</font></i></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:30"/>
( <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="#E7:30"><i><font color="Green" title="E26">a1</font></i></a>, <a class="txt" href="#E9:30"><i><font color="Green" title="E27">Z0</font></i></a>, <a class="txt" href="#E6:30"><i><font color="Green" title="E25">A0</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
