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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>;<br/>
<a NAME="E1:13_2_1"/><i><font color="Green">E10</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span> is <a href="lattice3.html#V1">with_suprema</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D10">LATTICE3:def 10</a><br/>

<a NAME="E1:13_2_1_1"/><i><font color="Green">E11</font></i>: 
<span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<b>thus </b><a NAME="E2:13_2_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> st <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_2_1_1_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 ) )
 </span><b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_4"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_4"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_4"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_4"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_4"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_5"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_5"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_5"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_5"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_5"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_5"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_6"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_6"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_6"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_6"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_6"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_6"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_6"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_6"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_7"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_7"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_7"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_7"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_7"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_7"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_7"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_7"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_8"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_8"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 )
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_1_1_8"/>
3 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = 3 as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E4:13_2_1_1_1_1_8"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_8_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_2_1_1_1_1_8_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_2_1_1_1_1_8_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:13_2_1_1_1_1_8_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_8"><i><font color="Green">E12</font></i></a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2_1_1_1_1_8_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:13_2_1_1_1_1_8_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:13_2_1_1_1_1_8_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_8"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:13_2_1_1_1_1_8_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:13_2_1_1_1_1_8"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><a NAME="E6:13_2_1_1_1_1_8"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_8"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><b>assume </b><a NAME="E8:13_2_1_1_1_1_8"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E9:13_2_1_1_1_1_8"/><b>then </b><i><font color="Green">E14</font></i>: 
( 1 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; 2 <a href="subset_1.html#K6">\</a> 1 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_8"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_8_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:13_2_1_1_1_1_8_2"/>
0 <a href="hidden.html#R2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_8_2"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_8"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_8_2"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_8_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_8_3"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:13_2_1_1_1_1_8_3"/>
1 <a href="hidden.html#R2">in</a> 2 <a href="subset_1.html#K6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:13_2_1_1_1_1_8_3"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_8"><i><font color="Green">E14</font></i></a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_8_3"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_8_3"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_8_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_8_4"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1
 ;<br/><a NAME="E2:13_2_1_1_1_1_8_4"/>
0 <a href="hidden.html#R2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_8_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_8"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_8_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_8_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_8_5"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2
 ;<br/><a NAME="E2:13_2_1_1_1_1_8_5"/>
0 <a href="hidden.html#R2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_8_5"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_8"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_8_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E14:13_2_1_1_1_1_8"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E7:13_2_1_1_1_1_8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow11.html#E10:13_2_1_1_1_1_8"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow11.html#E11:13_2_1_1_1_1_8"><i><font color="Green">E16</font></i></a>, <a class="txt" href="yellow11.html#E12:13_2_1_1_1_1_8"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_9"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_9"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 )
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_1_1_9"/>
3 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = 3 as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E4:13_2_1_1_1_1_9"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_9_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_2_1_1_1_1_9_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_9_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_2_1_1_1_1_9_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:13_2_1_1_1_1_9_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_9"><i><font color="Green">E12</font></i></a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2_1_1_1_1_9_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:13_2_1_1_1_1_9_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_9"><i><font color="Green">E12</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:13_2_1_1_1_1_9"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><a NAME="E6:13_2_1_1_1_1_9"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_9"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><b>assume </b><a NAME="E8:13_2_1_1_1_1_9"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E9:13_2_1_1_1_1_9"/><b>then </b><i><font color="Green">E14</font></i>: 
( 1 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; 3 <a href="subset_1.html#K6">\</a> 2 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_9"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_9_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_9_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:13_2_1_1_1_1_9_2"/>
0 <a href="hidden.html#R2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_9_2"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_9"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_9_2"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_9_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_9_3"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:13_2_1_1_1_1_9_3"/>
2 <a href="hidden.html#R2">in</a> 3 <a href="subset_1.html#K6">\</a> 2
 <b>by </b><i><a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_9_3"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_9"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_9_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_9_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_9_4"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1
 ;<br/><a NAME="E2:13_2_1_1_1_1_9_4"/>
0 <a href="hidden.html#R2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_9_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_9"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_9_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_9_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_9_5"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2
 ;<br/><a NAME="E2:13_2_1_1_1_1_9_5"/>
0 <a href="hidden.html#R2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_9_5"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_9"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_9_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E14:13_2_1_1_1_1_9"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E7:13_2_1_1_1_1_9"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow11.html#E10:13_2_1_1_1_1_9"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow11.html#E11:13_2_1_1_1_1_9"><i><font color="Green">E16</font></i></a>, <a class="txt" href="yellow11.html#E12:13_2_1_1_1_1_9"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_10"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_10"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_1_1_10"/>
1 <a href="tarski.html#R1">c=</a> 3
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_2_1_1_1_1_10_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 1
 ;<br/>

<a NAME="E2:13_2_1_1_1_1_10_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2_1_1_1_1_10_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 3
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_10"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E4:13_2_1_1_1_1_10"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E5:13_2_1_1_1_1_10"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E6:13_2_1_1_1_1_10"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E7:13_2_1_1_1_1_10"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E8:13_2_1_1_1_1_10"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E9:13_2_1_1_1_1_10"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_11"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_11"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_11"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_11"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_11"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_11"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_11"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_11"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_12"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_12"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_1_1_12"/>
3 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = 3 as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E4:13_2_1_1_1_1_12"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_12_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_2_1_1_1_1_12_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_12_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_2_1_1_1_1_12_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:13_2_1_1_1_1_12_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_12"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2_1_1_1_1_12_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:13_2_1_1_1_1_12_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:13_2_1_1_1_1_12_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_12"><i><font color="Green">E12</font></i></a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:13_2_1_1_1_1_12_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:13_2_1_1_1_1_12"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><a NAME="E6:13_2_1_1_1_1_12"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_12"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><b>assume </b><a NAME="E8:13_2_1_1_1_1_12"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E9:13_2_1_1_1_1_12"/><b>then </b><i><font color="Green">E14</font></i>: 
( 1 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; 2 <a href="subset_1.html#K6">\</a> 1 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_12"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_12_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:13_2_1_1_1_1_12_2"/>
0 <a href="hidden.html#R2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_12_2"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_12"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_12_2"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_12_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_12_3"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:13_2_1_1_1_1_12_3"/>
1 <a href="hidden.html#R2">in</a> 2 <a href="subset_1.html#K6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:13_2_1_1_1_1_12_3"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_12"><i><font color="Green">E14</font></i></a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_12_3"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_12_3"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_12_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_12_4"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1
 ;<br/><a NAME="E2:13_2_1_1_1_1_12_4"/>
0 <a href="hidden.html#R2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_12_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_12"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_12_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_12_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_12_5"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2
 ;<br/><a NAME="E2:13_2_1_1_1_1_12_5"/>
0 <a href="hidden.html#R2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_12_5"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_12"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_12_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E14:13_2_1_1_1_1_12"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E7:13_2_1_1_1_1_12"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow11.html#E10:13_2_1_1_1_1_12"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow11.html#E11:13_2_1_1_1_1_12"><i><font color="Green">E16</font></i></a>, <a class="txt" href="yellow11.html#E12:13_2_1_1_1_1_12"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_13"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_13"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_13"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_13"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_13"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_13"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_13"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_13"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_14"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_14"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 )
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_1_1_14"/>
3 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = 3 as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E4:13_2_1_1_1_1_14"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_14_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_2_1_1_1_1_14_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_14_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_2_1_1_1_1_14_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:13_2_1_1_1_1_14_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2_1_1_1_1_14_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:13_2_1_1_1_1_14_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:13_2_1_1_1_1_14_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:13_2_1_1_1_1_14_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:13_2_1_1_1_1_14"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><a NAME="E6:13_2_1_1_1_1_14"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_14"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><b>assume </b><a NAME="E8:13_2_1_1_1_1_14"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E9:13_2_1_1_1_1_14"/><b>then </b><i><font color="Green">E14</font></i>: 
( 2 <a href="subset_1.html#K6">\</a> 1 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; 3 <a href="subset_1.html#K6">\</a> 2 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_14_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_14_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:13_2_1_1_1_1_14_2"/>
1 <a href="hidden.html#R2">in</a> 2 <a href="subset_1.html#K6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_14_2"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_14"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_14_2"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_14_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_14_3"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:13_2_1_1_1_1_14_3"/>
1 <a href="hidden.html#R2">in</a> 2 <a href="subset_1.html#K6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:13_2_1_1_1_1_14_3"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_14"><i><font color="Green">E14</font></i></a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_14_3"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_14_3"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_14_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_14_4"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1
 ;<br/><a NAME="E2:13_2_1_1_1_1_14_4"/>
2 <a href="hidden.html#R2">in</a> 3 <a href="subset_1.html#K6">\</a> 2
 <b>by </b><i><a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_14_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_14"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_14_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_14_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_14_5"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2
 ;<br/><a NAME="E2:13_2_1_1_1_1_14_5"/>
1 <a href="hidden.html#R2">in</a> 2 <a href="subset_1.html#K6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_14_5"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_14"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_14_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E14:13_2_1_1_1_1_14"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E7:13_2_1_1_1_1_14"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow11.html#E10:13_2_1_1_1_1_14"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow11.html#E11:13_2_1_1_1_1_14"><i><font color="Green">E16</font></i></a>, <a class="txt" href="yellow11.html#E12:13_2_1_1_1_1_14"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_15"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_15"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_1_1_15"/>
2 <a href="subset_1.html#K6">\</a> 1 <a href="tarski.html#R1">c=</a> 3
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_2_1_1_1_1_15_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 2 <a href="subset_1.html#K6">\</a> 1
 ;<br/>

<a NAME="E2:13_2_1_1_1_1_15_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2_1_1_1_1_15_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 3
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_15"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E4:13_2_1_1_1_1_15"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E5:13_2_1_1_1_1_15"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E6:13_2_1_1_1_1_15"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E7:13_2_1_1_1_1_15"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E8:13_2_1_1_1_1_15"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E9:13_2_1_1_1_1_15"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_16"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_16"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_16"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_16"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_16"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_16"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_16"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_16"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_17"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_17"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_1_1_17"/>
3 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = 3 as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E4:13_2_1_1_1_1_17"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_17_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_2_1_1_1_1_17_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_17"><i><font color="Green">E12</font></i></a></i>;<br/>

<b>thus </b><a NAME="E2:13_2_1_1_1_1_17_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_17_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_2_1_1_1_1_17_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:13_2_1_1_1_1_17_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_17"><i><font color="Green">E12</font></i></a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2_1_1_1_1_17_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


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


</div><b>end;</b></div><b>hence </b><a NAME="E5:13_2_1_1_1_1_17"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><a NAME="E6:13_2_1_1_1_1_17"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_17"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><b>assume </b><a NAME="E8:13_2_1_1_1_1_17"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E9:13_2_1_1_1_1_17"/><b>then </b><i><font color="Green">E14</font></i>: 
( 1 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; 3 <a href="subset_1.html#K6">\</a> 2 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_17"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_17_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_17_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:13_2_1_1_1_1_17_2"/>
0 <a href="hidden.html#R2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_17_2"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_17"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_17_2"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_17_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_17_3"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:13_2_1_1_1_1_17_3"/>
2 <a href="hidden.html#R2">in</a> 3 <a href="subset_1.html#K6">\</a> 2
 <b>by </b><i><a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_17_3"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_17"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_17_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_17_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_17_4"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1
 ;<br/><a NAME="E2:13_2_1_1_1_1_17_4"/>
0 <a href="hidden.html#R2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_17_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_17"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_17_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_17_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_17_5"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2
 ;<br/><a NAME="E2:13_2_1_1_1_1_17_5"/>
0 <a href="hidden.html#R2">in</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_17_5"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_17"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_17_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E14:13_2_1_1_1_1_17"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E7:13_2_1_1_1_1_17"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow11.html#E10:13_2_1_1_1_1_17"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow11.html#E11:13_2_1_1_1_1_17"><i><font color="Green">E16</font></i></a>, <a class="txt" href="yellow11.html#E12:13_2_1_1_1_1_17"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_18"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_18"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 )
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_1_1_18"/>
3 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = 3 as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E4:13_2_1_1_1_1_18"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_18_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_2_1_1_1_1_18_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:13_2_1_1_1_1_18_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:13_2_1_1_1_1_18_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:13_2_1_1_1_1_18_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:13_2_1_1_1_1_18"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><a NAME="E6:13_2_1_1_1_1_18"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_18"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><b>assume </b><a NAME="E8:13_2_1_1_1_1_18"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E9:13_2_1_1_1_1_18"/><b>then </b><i><font color="Green">E14</font></i>: 
( 2 <a href="subset_1.html#K6">\</a> 1 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; 3 <a href="subset_1.html#K6">\</a> 2 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_18_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_18_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:13_2_1_1_1_1_18_2"/>
1 <a href="hidden.html#R2">in</a> 2 <a href="subset_1.html#K6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_18_2"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_18_2"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_18_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_18_3"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:13_2_1_1_1_1_18_3"/>
1 <a href="hidden.html#R2">in</a> 2 <a href="subset_1.html#K6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:13_2_1_1_1_1_18_3"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_18"><i><font color="Green">E14</font></i></a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_18_3"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_18_3"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_18_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_18_4"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1
 ;<br/><a NAME="E2:13_2_1_1_1_1_18_4"/>
2 <a href="hidden.html#R2">in</a> 3 <a href="subset_1.html#K6">\</a> 2
 <b>by </b><i><a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_18_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_18_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_18_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1_1_1_18_5"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2
 ;<br/><a NAME="E2:13_2_1_1_1_1_18_5"/>
1 <a href="hidden.html#R2">in</a> 2 <a href="subset_1.html#K6">\</a> 1
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1_1_1_18_5"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E9:13_2_1_1_1_1_18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_18_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E14:13_2_1_1_1_1_18"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E7:13_2_1_1_1_1_18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow11.html#E10:13_2_1_1_1_1_18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow11.html#E11:13_2_1_1_1_1_18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="yellow11.html#E12:13_2_1_1_1_1_18"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_19"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_19"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_19"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_19"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_19"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_19"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_19"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_19"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_20"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_20"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_20"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_20"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_20"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_20"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_20"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_20"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_20"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_21"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_21"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_21"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_21"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_21"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_21"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_21"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_21"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_22"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_22"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_1_1_22"/>
1 <a href="tarski.html#R1">c=</a> 3
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_2_1_1_1_1_22_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 1
 ;<br/>

<a NAME="E2:13_2_1_1_1_1_22_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2_1_1_1_1_22_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 3
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_22"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E4:13_2_1_1_1_1_22"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E5:13_2_1_1_1_1_22"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E6:13_2_1_1_1_1_22"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E7:13_2_1_1_1_1_22"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E8:13_2_1_1_1_1_22"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E9:13_2_1_1_1_1_22"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_23"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_23"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 )
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_1_1_23"/>
2 <a href="subset_1.html#K6">\</a> 1 <a href="tarski.html#R1">c=</a> 3
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1_1_23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_2_1_1_1_1_23_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 2 <a href="subset_1.html#K6">\</a> 1
 ;<br/>

<a NAME="E2:13_2_1_1_1_1_23_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2_1_1_1_1_23_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 3
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_23"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E4:13_2_1_1_1_1_23"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E5:13_2_1_1_1_1_23"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E6:13_2_1_1_1_1_23"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E7:13_2_1_1_1_1_23"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E8:13_2_1_1_1_1_23"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E9:13_2_1_1_1_1_23"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_24"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_24"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1_1_1_1_24"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_24"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_24"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_24"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_24"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_24"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_24"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1_1_25"><b>suppose </b></a><a NAME="E1:13_2_1_1_1_1_25"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><a NAME="E3:13_2_1_1_1_1_25"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1_1_25"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p3">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p3">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span></span>)</span>;<br/><b>assume </b><a NAME="E5:13_2_1_1_1_1_25"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:13_2_1_1_1_1_25"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E7:13_2_1_1_1_1_25"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E8:13_2_1_1_1_1_25"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div>
<a NAME="E2:13_2_1"/>
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span> is <a href="lattice3.html#V2">with_infima</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_2_1_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span> )
 ;<br/><a NAME="E2:13_2_1_2_1"/><b>then </b><i><font color="Green">E11</font></i>: 
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 ) &amp; ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 ) )
 <b>by </b><i><a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><a NAME="E3:13_2_1_2_1"/>
1 <a href="xboole_0.html#R1">misses</a> 2 <a href="subset_1.html#K6">\</a> 1
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E4:13_2_1_2_1"/><b>then </b><i><font color="Green">E12</font></i>: 
1 <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:13_2_1_2_1"/><i><font color="Green">E13</font></i>: 
1 <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_2_1_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 1 <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>
 ;<br/><a NAME="E2:13_2_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 3 <a href="subset_1.html#K6">\</a> 2 )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:13_2_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 <b>by </b><i><a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_2_1_2_1_1"/>
1 <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:13_2_1_2_1"/>
1 <a href="tarski.html#R1">c=</a> 3
 <b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/><a NAME="E7:13_2_1_2_1"/><b>then </b><i><font color="Green">E14</font></i>: 
1 <a href="xboole_0.html#K3">/\</a> 3 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E8:13_2_1_2_1"/><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_2_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_2_1_2_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>
 ;<br/><a NAME="E2:13_2_1_2_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 2 <a href="subset_1.html#K6">\</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 3 <a href="subset_1.html#K6">\</a> 2 )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:13_2_1_2_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 <b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="yellow11.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_2_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_2_1_2_1_2"/>
<span class="p1">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:13_2_1_2_1"/>
2 <a href="subset_1.html#K6">\</a> 1 <a href="tarski.html#R1">c=</a> 3
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_2_1_2_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 2 <a href="subset_1.html#K6">\</a> 1
 ;<br/>

<a NAME="E2:13_2_1_2_1_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2_1_2_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> 3
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:13_2_1_2_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span> <a href="subset_1.html#K8">/\</a> 3 <a href="hidden.html#R1">=</a> 2 <a href="subset_1.html#K6">\</a> 1
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E11:13_2_1_2_1"/>
<span class="p1">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span> <a href="subset_1.html#K8">/\</a> 3 <a href="hidden.html#R1">=</a> 3 <a href="subset_1.html#K6">\</a> 2
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E12:13_2_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span>
 <b>by </b><i><a class="txt" href="yellow11.html#E2:13_2_1_2_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="yellow11.html#E4:13_2_1_2_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow11.html#E5:13_2_1_2_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow11.html#E7:13_2_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E8:13_2_1_2_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow11.html#E10:13_2_1_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_2_1_2"/>
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,<span class="p2">(<span class="default">2 <a href="subset_1.html#K6">\</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="subset_1.html#K6">\</a> 2</span>)</span>,3</span><a href="enumset1.html#K3">}</a></span> is <a href="lattice3.html#V2">with_infima</a>
 <b>by </b><i><a class="ref" href="yellow_1.html#T12">YELLOW_1:12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_2_1"/>
(  <a href="yellow11.html#K2" target="_self">M_3</a>  is <a href="lattice3.html#V2">with_infima</a> &amp;  <a href="yellow11.html#K2" target="_self">M_3</a>  is <a href="lattice3.html#V1">with_suprema</a> )
 <b>by </b><i><a class="txt" href="yellow11.html#E1:13_2_1"><i><font color="Green">E10</font></i></a></i>;<br/>


</div>
