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

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="dilworth.html#V4" title="DILWORTH:attr.4">with_finite_stability#</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:101_1"/>
( <font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  or  not <font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1"/>
<font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:101_1_1"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T51" target="_self" title="DILWORTH:th.51">DWf</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_2"/><span class="lab"><font color="Green" title="E56">S0</font></span>: 
not <font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font></span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">P</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="dilworth.html#V4" title="DILWORTH:attr.4">with_finite_stability#</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> ex <font color="Olive" title="b2">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Olive" title="b1">P</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Olive" title="b1">P</font>;<br/><a NAME="E2:101_1_2"/><span class="lab"><font color="Green" title="E57">P1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/><a NAME="E3:101_1_2"/><span class="lab"><font color="Green" title="E58">P2</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_2_1"/><span class="lab"><font color="Green" title="E59">A</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">P</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="dilworth.html#V4" title="DILWORTH:attr.4">with_finite_stability#</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c3">P</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:101_1_2_1"/><span class="lab"><font color="Green" title="E60">B</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c3">P</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">C</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">P</font><span class="kw"> such that </span><br/><a NAME="E4:101_1_2_1"/><span class="lab"><font color="Green" title="E61">C</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="dilworth.html#V7" title="DILWORTH:attr.7">strong-chain</a> 
 <span class="kw">and </span><br/><a NAME="E5:101_1_2_1"/><span class="lab"><font color="Green" title="E62">D</font></span>: 
for <font color="Olive" title="b1">D</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">P</font>  holds <br/>(  not <font color="Olive" title="b1">D</font> is  <a href="dilworth.html#V7" title="DILWORTH:attr.7">strong-chain</a>  or  not <font color="Maroon" title="c4">C</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b1">D</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T52" target="_self" title="DILWORTH:th.52">Maxsc</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">cP</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:101_1_2_1_1"/>
( <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> or <font color="Maroon" title="c4">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_2_1_1_1"/><span class="lab"><font color="Green" title="E63">S1</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c3">P</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font></span><br/><div class="add"><a NAME="E2:101_1_2_1_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">C</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Olive" title="b1">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_2_1"><span class="lab"><font color="Green" title="E61">C</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">Cp</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">C</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c3">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D11" target="_self" title="DILWORTH:def.11">LCpart</a>, <a class="txt" href="#E1:101_1_2_1_1_1"><span class="lab"><font color="Green" title="E63">S1</font></span></a>, <a class="ref" href="eqrel_1.html#T48" title="EQREL_1:th.48">EQREL_1:48</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">Cp</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">Cp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font></span><br/><a NAME="E4:101_1_2_1_1_1"/><span class="lab"><font color="Green" title="E64">R</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">P</font>
 ;<br/><span class="kw">thus </span> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">Cp</font> = 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></span>
<br/>.= 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">R</font></span></a>, <a class="txt" href="#E4:101_1_2_1"><span class="lab"><font color="Green" title="E61">C</font></span></a>, <a class="txt" href="#E1:101_1_2_1_1_1"><span class="lab"><font color="Green" title="E63">S1</font></span></a>, <a class="ref" href="dilworth.html#T20" target="_self" title="DILWORTH:th.20">Width3</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_2_1_1_2"/>
<font color="Maroon" title="c4">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c3">P</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font></span><br/><div class="add"><a NAME="E2:101_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">S1a</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">cP9</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">C</font>;<br/><span class="lab"><font color="Green" title="E64">E</font></span>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">C</font></span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T48" title="XBOOLE_1:th.48">XBOOLE_1:48</a></span>
<br/>.= 
<font color="Maroon" title="c4">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">cP9</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">P</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c8">P9</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">cP9</font>;<br/><a NAME="E5:101_1_2_1_1_2"/>
<font color="Maroon" title="c7">cP9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E63">S1a</font></span></a>, <a class="ref" href="xboole_1.html#T105" title="XBOOLE_1:th.105">XBOOLE_1:105</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">P9</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">cP9</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="dilworth.html#V4" title="DILWORTH:attr.4">with_finite_stability#</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><a NAME="E7:101_1_2_1_1_2"/><span class="lab"><font color="Green" title="E65">cP9c</font></span>: 
<font color="Maroon" title="c7">cP9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">P9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/><a NAME="E8:101_1_2_1_1_2"/><span class="lab"><font color="Green" title="E66">G</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c9">P9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_2_1"><span class="lab"><font color="Green" title="E60">B</font></span></a>, <a class="ref" href="dilworth.html#T44" target="_self" title="DILWORTH:th.44">Wsubp0</a></span>;<br/><a NAME="E9:101_1_2_1_1_2"/><span class="lab"><font color="Green" title="E67">H</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c9">P9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E68">AA</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c9">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">A</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c9">P9</font><span class="kw"> such that </span><br/><a NAME="E3:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E69">A1</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c9">P9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D6" target="_self" title="DILWORTH:def.6">Lwidth</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">A9</font> = <font color="Maroon" title="c10">A</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c3">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="dilworth.html#T30" target="_self" title="DILWORTH:th.30">SPAChain1</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 or  not $1 <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> $3 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> or  ex <font color="Olive" title="b2">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; $3 <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">P</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">p</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> $1</span>)</span>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[$1,<font color="Olive" title="b1">p</font>,$2];<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">S</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">P</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S</font> &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">$1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>] );<br/>
<a NAME="E5:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E70">P</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c12">a</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_2_1"/><span class="lab"><font color="Green" title="E71">A2</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c12">a</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E2:101_1_2_1_1_2_2_1"/>
<font color="Maroon" title="c10">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">cP9c</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E3:101_1_2_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">Ca</font> = <font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E72">A3</font></span>: 
<font color="Maroon" title="c13">Ca</font> is  <a href="dilworth.html#V7" title="DILWORTH:attr.7">strong-chain</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:101_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:101_1_2_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">B3</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Ca</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E4:101_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E74">C3</font></span>: 
not <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">cP9c</font></span></a>, <a class="txt" href="#E1:101_1_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E5:101_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">Ca</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E6:101_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c13">Ca</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E73">B3</font></span></a>, <a class="txt" href="#E4:101_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E74">C3</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E7:101_1_2_1_1_2_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="txt" href="#E5:101_1_2_1"><span class="lab"><font color="Green" title="E62">D</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">S</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">P</font><span class="kw"> such that </span><br/><a NAME="E7:101_1_2_1_1_2_2_1"/><span class="lab"><font color="Green" title="E72">B2</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">S</font>,<font color="Maroon" title="c13">Ca</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D13" target="_self" title="DILWORTH:def.13">Lsc</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">S</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c14">S</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c14">S</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">S</font>,<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>] )</span><br/>

<span class="kw">thus </span><a NAME="E8:101_1_2_1_1_2_2_1"/>
<font color="Maroon" title="c14">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">S</font>,<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>] )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E73">A3</font></span>: 
not <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:101_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E74">A3b</font></span>: 
<font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">Ca</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">Ca</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_2_1_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">Ca</font></span><br/>

<a NAME="E2:101_1_2_1_1_2_2_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:101_1_2_1_1_2_2_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Ca</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:101_1_2_1_1_2_2_1_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">Ca</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:101_1_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">Ca</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">Ca</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_2_1_2_2"/><span class="lab"><font color="Green" title="E75">A4</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">Ca</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font></span><br/>

<a NAME="E2:101_1_2_1_1_2_2_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">B4</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:101_1_2_1_1_2_2_1_2_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Ca</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_2_1_2_2"><span class="lab"><font color="Green" title="E75">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:101_1_2_1_1_2_2_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:101_1_2_1_1_2_2_1_2_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_2_1_1_2_2_1_2_2"><span class="lab"><font color="Green" title="E76">B4</font></span></a>, <a class="txt" href="#E1:101_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:101_1_2_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A3a</font></span>: 
<font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">Ca</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E74">A3b</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">p</font> being   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c14">S</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:101_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E76">B3</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E7:101_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E77">C3</font></span>: 
 not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c14">S</font>,<font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c4">C</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_2_1"><span class="lab"><font color="Green" title="E61">C</font></span></a>, <a class="ref" href="dilworth.html#D13" target="_self" title="DILWORTH:def.13">Lsc</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:101_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E78">D3</font></span>: 
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font>
 <span class="kw">and </span><br/><a NAME="E10:101_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E79">E3</font></span>: 
<font color="Maroon" title="c14">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">c</font>
 <span class="kw">and </span><br/><a NAME="E11:101_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E80">F3</font></span>: 
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">c</font> holds <br/><font color="Maroon" title="c4">C</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E77">C3</font></span></a></span>;<br/><a NAME="E12:101_1_2_1_1_2_2_1_2"/>
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">c</font> holds <br/><font color="Maroon" title="c13">Ca</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font> &amp; <font color="Maroon" title="c17">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">c</font> implies <font color="Maroon" title="c13">Ca</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:101_1_2_1_1_2_2_1_2_3"/><span class="lab"><font color="Green" title="E81">A4</font></span>: 
<font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font>
 <span class="kw">and </span><br/><a NAME="E2:101_1_2_1_1_2_2_1_2_3"/><span class="lab"><font color="Green" title="E82">B4</font></span>: 
<font color="Maroon" title="c17">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">Ca</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2_1_2_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_2_1_2_3_1"/>
<font color="Maroon" title="c13">Ca</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_2_1_1_2_2_1_2_3_1"/><span class="lab"><font color="Green" title="E83">A5</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Ca</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c17">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E4:101_1_2_1_1_2_2_1_2_3_1"/><span class="lab"><font color="Green" title="E84">B5</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Ca</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2_1_1_2_2_1_2_3_1"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:101_1_2_1_1_2_2_1_2_3_1"/><span class="lab"><font color="Green" title="E85">C5</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2_1_1_2_2_1_2_3_1"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2_2_1_2_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:101_1_2_1_1_2_2_1_2_3_1_1"/>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_2_1_1_2_2_1_2_3_1"><span class="lab"><font color="Green" title="E84">B5</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2_2_1_2_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_2_1_1_2_2_1_2_3_1_1_1"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:101_1_2_1_1_2_2_1_2_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c17">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_1_2_1_1_2_2_1_2_3_1"><span class="lab"><font color="Green" title="E85">C5</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:101_1_2_1_1_2_2_1_2_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_2_1_2_3"><span class="lab"><font color="Green" title="E81">A4</font></span></a>, <a class="txt" href="#E2:101_1_2_1_1_2_2_1_2_3"><span class="lab"><font color="Green" title="E82">B4</font></span></a>, <a class="txt" href="#E11:101_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E80">F3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2_2_1_2_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_2_1_1_2_2_1_2_3_1_1_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:101_1_2_1_1_2_2_1_2_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:101_1_2_1_1_2_2_1_2_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c14">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_1_2_1_1_2_2_1_2_3_1"><span class="lab"><font color="Green" title="E85">C5</font></span></a>, <a class="txt" href="#E1:101_1_2_1_1_2_2_1_2_3"><span class="lab"><font color="Green" title="E81">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:101_1_2_1_1_2_2_1_2_3_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:101_1_2_1_1_2_2_1_2_3"/>
<font color="Maroon" title="c13">Ca</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:101_1_2_1_1_2_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:101_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E78">D3</font></span></a>, <a class="txt" href="#E10:101_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E79">E3</font></span></a>, <a class="txt" href="#E6:101_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E76">B3</font></span></a>, <a class="txt" href="#E4:101_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E75">A3a</font></span></a>, <a class="txt" href="#E7:101_1_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E72">B2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:101_1_2_1_1_2_2_1"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">S</font>,<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/>

<span class="kw">thus </span><a NAME="E11:101_1_2_1_1_2_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">S</font>,<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E72">B2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c12">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E7:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E71">C1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A</font>
 <span class="kw">and </span><br/><a NAME="E8:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E72">E1</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="#E5:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E70">P</font></span></a>);<br/></span>
<a NAME="E9:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E73">D1a</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E71">C1</font></span></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">SS</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font></span>)</span>;<br/>
<a NAME="E10:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E74">C1bb</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> holds <br/><font color="Olive" title="b1">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> implies <font color="Maroon" title="c14">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_2_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_2_1_1_2_2_2"/><span class="lab"><font color="Green" title="E75">A2</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">and </span><br/><a NAME="E4:101_1_2_1_1_2_2_2"/><span class="lab"><font color="Green" title="E76">B2</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">S</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">P</font><span class="kw"> such that </span><br/><a NAME="E6:101_1_2_1_1_2_2_2"/><span class="lab"><font color="Green" title="E77">C2</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">S</font>
 <span class="kw">and </span><br/><a NAME="E7:101_1_2_1_1_2_2_2"/>
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">S</font>,<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E71">C1</font></span></a>, <a class="txt" href="#E3:101_1_2_1_1_2_2_2"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="txt" href="#E8:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E72">E1</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:101_1_2_1_1_2_2_2"/>
<font color="Maroon" title="c14">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101_1_2_1_1_2_2_2"><span class="lab"><font color="Green" title="E77">C2</font></span></a>, <a class="txt" href="#E4:101_1_2_1_1_2_2_2"><span class="lab"><font color="Green" title="E76">B2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E75">C1c</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font></span>)</span> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_2_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_2_1_1_2_2_3"/><span class="lab"><font color="Green" title="E76">A2</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">y</font>
 <span class="kw">and </span><br/><a NAME="E4:101_1_2_1_1_2_2_3"/><span class="lab"><font color="Green" title="E77">B2</font></span>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:101_1_2_1_1_2_2_3"/><span class="lab"><font color="Green" title="E78">C2</font></span>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">and </span><br/><a NAME="E7:101_1_2_1_1_2_2_3"/><span class="lab"><font color="Green" title="E79">D2</font></span>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_2_1_1_2_2_3"><span class="lab"><font color="Green" title="E77">B2</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">S</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">P</font><span class="kw"> such that </span><br/><a NAME="E9:101_1_2_1_1_2_2_3"/><span class="lab"><font color="Green" title="E80">E2</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">S</font>
 <span class="kw">and </span><br/><a NAME="E10:101_1_2_1_1_2_2_3"/>
( <font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c17">S</font>,<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E71">C1</font></span></a>, <a class="txt" href="#E6:101_1_2_1_1_2_2_3"><span class="lab"><font color="Green" title="E78">C2</font></span></a>, <a class="txt" href="#E8:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E72">E1</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:101_1_2_1_1_2_2_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2_1_1_2_2_3"><span class="lab"><font color="Green" title="E76">A2</font></span></a>, <a class="txt" href="#E7:101_1_2_1_1_2_2_3"><span class="lab"><font color="Green" title="E79">D2</font></span></a>, <a class="txt" href="#E9:101_1_2_1_1_2_2_3"><span class="lab"><font color="Green" title="E80">E2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E76">C1a</font></span>: 
<font color="Maroon" title="c11">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A9</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_2_4"/><span class="lab"><font color="Green" title="E77">A2</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">S</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">P</font><span class="kw"> such that </span><br/><a NAME="E3:101_1_2_1_1_2_2_4"/>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">S</font>
 <span class="kw">and </span><br/><a NAME="E4:101_1_2_1_1_2_2_4"/><span class="lab"><font color="Green" title="E78">D2</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">and </span><br/><a NAME="E5:101_1_2_1_1_2_2_4"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c15">S</font>,<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E72">E1</font></span></a></span>;<br/>
<a NAME="E6:101_1_2_1_1_2_2_4"/>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_2_4"><span class="lab"><font color="Green" title="E77">A2</font></span></a>, <a class="txt" href="#E7:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E71">C1</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:101_1_2_1_1_2_2_4"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_2_1_1_2_2_4"><span class="lab"><font color="Green" title="E78">D2</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">SS</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font></span>)</span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E74">C1bb</font></span></a>, <a class="txt" href="#E9:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E73">D1a</font></span></a>, <a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">FINSET_1:25</a>, <a class="txt" href="#E11:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E75">C1c</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">SSp</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c14">SS</font>;<br/>
<a NAME="E14:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E77">cSS</font></span>: 
<font color="Maroon" title="c14">SS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c14">SS</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">p</font> being   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c14">SS</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E16:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E78">F1</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E17:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E79">G1</font></span>: 
 not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c14">SS</font>,<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c4">C</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_2_1"><span class="lab"><font color="Green" title="E61">C</font></span></a>, <a class="ref" href="dilworth.html#D13" target="_self" title="DILWORTH:def.13">Lsc</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E19:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E80">H1</font></span>: 
<font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">p</font>
 <span class="kw">and </span><br/><a NAME="E20:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E81">I1</font></span>: 
<font color="Maroon" title="c14">SS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">c</font>
 <span class="kw">and </span><br/><a NAME="E21:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E82">J1</font></span>: 
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">p</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">c</font> holds <br/><font color="Maroon" title="c4">C</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E79">G1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">c</font> = <font color="Maroon" title="c17">c</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c16">p</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E80">H1</font></span></a></span>;<br/>
<a NAME="E23:101_1_2_1_1_2_2"/>
<font color="Maroon" title="c11">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A9</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">SS</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E76">C1a</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">A</font> = <font color="Maroon" title="c11">A9</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c14">SS</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T31" target="_self" title="DILWORTH:th.31">SPAChain</a></span>;<br/>
<a NAME="E25:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E83">K1</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c14">SS</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c19">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D6" target="_self" title="DILWORTH:def.6">Lwidth</a></span>;<br/>
<a NAME="E26:101_1_2_1_1_2_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c14">SS</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T46" target="_self" title="DILWORTH:th.46">Chpw</a></span>;<br/>
<a NAME="E27:101_1_2_1_1_2_2"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c19">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E83">K1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c19">A</font><span class="kw"> such that </span><br/><a NAME="E29:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E84">L1</font></span>: 
<font color="Maroon" title="c18">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c19">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T48" target="_self" title="DILWORTH:th.48">ACpart2</a>, <a class="txt" href="#E16:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E78">F1</font></span></a>, <a class="txt" href="#E3:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E1:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E68">AA</font></span></a>, <a class="txt" href="#E2:101_1_2_1"><span class="lab"><font color="Green" title="E60">B</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E30:101_1_2_1_1_2_2"/>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c19">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E84">L1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E31:101_1_2_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">M1</font></span>: 
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">S</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">P</font><span class="kw"> such that </span><br/><a NAME="E33:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E86">N1</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">S</font>
 <span class="kw">and </span><br/><a NAME="E34:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E87">O1</font></span>: 
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
 <span class="kw">and </span><br/><a NAME="E35:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E88">P1</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c21">S</font>,<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E72">E1</font></span></a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> = $1 <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c21">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">Sp</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">e</font>) where <font color="Olive" title="b1">e</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c16">p</font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">e</font>] </span>}</span> ;<br/>
<a NAME="E36:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E89">R1</font></span>: 
<font color="Maroon" title="c21">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">SS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">S</font> or <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">SS</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_2_5"/><span class="lab"><font color="Green" title="E90">A2</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">SS</font></span><br/>

<a NAME="E2:101_1_2_1_1_2_2_5"/>
<font color="Maroon" title="c21">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E86">N1</font></span></a>, <a class="txt" href="#E7:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E71">C1</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:101_1_2_1_1_2_2_5"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">SS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_2_5"><span class="lab"><font color="Green" title="E90">A2</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">Sp</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">e</font>) where <font color="Olive" title="b1">e</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c16">p</font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">e</font>] </span>}</span>  as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c21">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E77">cSS</font></span></a>, <a class="ref" href="pua2mss1.html#T11" title="PUA2MSS1:th.11">PUA2MSS1:11</a></span>;<br/>
<a NAME="E38:101_1_2_1_1_2_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Sp</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c21">S</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Sp</font> implies <font color="Maroon" title="c24">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c21">S</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_2_6"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Sp</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c21">S</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">e</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c16">p</font><span class="kw"> such that </span><br/><a NAME="E3:101_1_2_1_1_2_2_6"/><span class="lab"><font color="Green" title="E90">A2</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">e</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font>
 <span class="kw">and </span><br/><a NAME="E4:101_1_2_1_1_2_2_6"/>
<font color="Maroon" title="c25">e</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c21">S</font>
 ;<br/>
<a NAME="E5:101_1_2_1_1_2_2_6"/>
<font color="Maroon" title="c25">e</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c14">SS</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D11" target="_self" title="DILWORTH:def.11">LCpart</a></span>;<br/>
<a NAME="E6:101_1_2_1_1_2_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c25">e</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T28" target="_self" title="DILWORTH:th.28">SPClique</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:101_1_2_1_1_2_2_6"/>
<font color="Maroon" title="c24">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c21">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2_1_1_2_2_6"><span class="lab"><font color="Green" title="E90">A2</font></span></a>, <a class="ref" href="dilworth.html#T29" target="_self" title="DILWORTH:th.29">SPClique1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">Sp</font> = <font color="Maroon" title="c23">Sp</font> as   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c21">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a>, <a class="ref" href="dilworth.html#D11" target="_self" title="DILWORTH:def.11">LCpart</a></span>;<br/>
<a NAME="E40:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E90">T1a</font></span>: 
<font color="Maroon" title="c24">Sp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">e</font>) where <font color="Olive" title="b1">e</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c16">p</font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">e</font>] </span>}</span> 
 
;<br/>
<a NAME="E41:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E91">T1b</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c24">Sp</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">p</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="dilworth.html#S1" target="_self" title="DILWORTH:sch.1">DILWORTH:sch 1</a>(<a class="txt" href="#E40:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E90">T1a</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c25">cc</font> = <font color="Maroon" title="c18">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font>;<br/>
<a NAME="E42:101_1_2_1_1_2_2"/>
<font color="Maroon" title="c18">c</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c21">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E85">M1</font></span></a>, <a class="txt" href="#E33:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E86">N1</font></span></a>, <a class="txt" href="#E34:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E87">O1</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E43:101_1_2_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">U1</font></span>: 
<font color="Maroon" title="c18">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Sp</font>
 
;<br/>
<a NAME="E44:101_1_2_1_1_2_2"/>
<font color="Maroon" title="c21">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> or <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_2_7"/><span class="lab"><font color="Green" title="E93">A2</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font></span><br/>

<a NAME="E2:101_1_2_1_1_2_2_7"/><span class="lab"><font color="Green" title="E94">B2</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_2_7"><span class="lab"><font color="Green" title="E93">A2</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:101_1_2_1_1_2_2_7"/><span class="lab"><font color="Green" title="E95">C2</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_2_7"><span class="lab"><font color="Green" title="E93">A2</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2_2_7_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:101_1_2_1_1_2_2_7_1"/>
( <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> or <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2_1_1_2_2_7"><span class="lab"><font color="Green" title="E95">C2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2_2_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_2_1_1_2_2_7_1_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font></span><br/><div class="add"><a NAME="E2:101_1_2_1_1_2_2_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">SS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_2_1_1_2_2_7"><span class="lab"><font color="Green" title="E94">B2</font></span></a>, <a class="txt" href="#E36:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E89">R1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:101_1_2_1_1_2_2_7_1_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_2_1_1_2_2_7"><span class="lab"><font color="Green" title="E94">B2</font></span></a>, <a class="txt" href="#E20:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E81">I1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2_2_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_2_1_1_2_2_7_1_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font></span><br/><div class="add"><a NAME="E2:101_1_2_1_1_2_2_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:101_1_2_1_1_2_2_7_1_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E85">M1</font></span></a>, <a class="txt" href="#E33:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E86">N1</font></span></a>, <a class="txt" href="#E34:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E87">O1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E46:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E93">V1</font></span>: 
<font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Sp</font>
 <span class="kw">and </span><br/><a NAME="E47:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E94">W1</font></span>: 
<font color="Maroon" title="c26">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font>
 <span class="kw">and </span><br/><a NAME="E48:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E95">X1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c26">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E92">U1</font></span></a>, <a class="txt" href="#E41:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E91">T1b</font></span></a>, <a class="txt" href="#E16:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E78">F1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="#E35:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E88">P1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">dd</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c16">p</font><span class="kw"> such that </span><br/><a NAME="E50:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E96">Y1</font></span>: 
<font color="Maroon" title="c26">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">dd</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font>
 <span class="kw">and </span><br/><a NAME="E51:101_1_2_1_1_2_2"/>
<font color="Maroon" title="c27">dd</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c21">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E93">V1</font></span></a></span>;<br/>
<a NAME="E52:101_1_2_1_1_2_2"/>
<font color="Maroon" title="c4">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c27">dd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E96">Y1</font></span></a>, <a class="txt" href="#E47:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E94">W1</font></span></a>, <a class="txt" href="#E21:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E82">J1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E97">Z1a</font></span>: <font color="Maroon" title="c4">C</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c26">d</font> = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E96">Y1</font></span></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 

;<br/>
<a NAME="E54:101_1_2_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c26">d</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c26">d</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E95">X1</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E56:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E98">Y1a</font></span>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c26">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E97">Z1a</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E57:101_1_2_1_1_2_2"/>
( <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E98">Y1a</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E58:101_1_2_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E59:101_1_2_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">dd</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E96">Y1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E60:101_1_2_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">c</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c27">dd</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E85">M1</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E61:101_1_2_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E96">Y1</font></span></a>, <a class="txt" href="#E47:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E94">W1</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E10:101_1_2_1_1_2"/><span class="kw">then </span>
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c9">P9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">G</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E11:101_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">Ha</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c9">P9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">A</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c3">P</font><span class="kw"> such that </span><br/><a NAME="E13:101_1_2_1_1_2"/><span class="lab"><font color="Green" title="E69">I</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D6" target="_self" title="DILWORTH:def.6">Lwidth</a></span>;<br/><a NAME="E14:101_1_2_1_1_2"/><span class="lab"><font color="Green" title="E70">Y</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c9">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:101_1_2_1_1_2_3_1"/>
( <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_2_1_1_2_3_1_1"/><span class="lab"><font color="Green" title="E71">S2</font></span>: 
<font color="Maroon" title="c10">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c9">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font></span><br/><div class="add"><a NAME="E2:101_1_2_1_1_2_3_1_1"/>
<font color="Maroon" title="c10">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">cP9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_3_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">cP9</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_3_1_1_1"/><span class="lab"><font color="Green" title="E72">A3</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">cP9</font></span><br/>

<a NAME="E2:101_1_2_1_1_2_3_1_1_1"/>
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_3_1_1_1"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="txt" href="#E1:101_1_2_1_1_2_3_1_1"><span class="lab"><font color="Green" title="E71">S2</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:101_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">cP9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_3_1_1_1"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:101_1_2_1_1_2_3_1_1"/>
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c9">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_2_1"><span class="lab"><font color="Green" title="E60">B</font></span></a>, <a class="txt" href="#E9:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">H</font></span></a>, <a class="txt" href="#E13:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">I</font></span></a>, <a class="ref" href="dilworth.html#T45" target="_self" title="DILWORTH:th.45">Wsubp1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_2_1_1_2_3_1_2"/>
<font color="Maroon" title="c10">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c9">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_2_1_1_2_3_1_2"/><span class="lab"><font color="Green" title="E71">A3</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E4:101_1_2_1_1_2_3_1_2"/><span class="lab"><font color="Green" title="E72">B3</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:101_1_2_1_1_2_3_1_2"/><span class="lab"><font color="Green" title="E73">C3</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">A9</font> = <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><a NAME="E6:101_1_2_1_1_2_3_1_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_2_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E72">B3</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><a NAME="E7:101_1_2_1_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">D3</font></span>: 
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/><a NAME="E8:101_1_2_1_1_2_3_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E9:101_1_2_1_1_2_3_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E10:101_1_2_1_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">D4a</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_2_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E74">D3</font></span></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></span>;<br/><a NAME="E11:101_1_2_1_1_2_3_1_2"/><span class="lab"><font color="Green" title="E76">E4</font></span>: 
<font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/><a NAME="E12:101_1_2_1_1_2_3_1_2"/>
<font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">cP9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_3_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">xx</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">cP9</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_3_1_2_1"/><span class="lab"><font color="Green" title="E77">A5</font></span>: 
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">cP9</font></span><br/>

<a NAME="E2:101_1_2_1_1_2_3_1_2_1"/><span class="lab"><font color="Green" title="E78">B5</font></span>: 
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_3_1_2_1"><span class="lab"><font color="Green" title="E77">A5</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E3:101_1_2_1_1_2_3_1_2_1"/><span class="lab"><font color="Green" title="E79">C5</font></span>: 
not <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_3_1_2_1"><span class="lab"><font color="Green" title="E77">A5</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:101_1_2_1_1_2_3_1_2_1"/>
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2_1_1_2_3_1_2_1"><span class="lab"><font color="Green" title="E79">C5</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:101_1_2_1_1_2_3_1_2_1"/><span class="kw">then </span>
not <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_2_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E72">B3</font></span></a>, <a class="txt" href="#E5:101_1_2_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E73">C3</font></span></a>, <a class="txt" href="#E2:101_1_2_1_1_2_3_1_2_1"><span class="lab"><font color="Green" title="E78">B5</font></span></a>, <a class="txt" href="#E4:101_1_2_1"><span class="lab"><font color="Green" title="E61">C</font></span></a>, <a class="ref" href="dilworth.html#T15" target="_self" title="DILWORTH:th.15">ACClique</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:101_1_2_1_1_2_3_1_2_1"/>
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">cP9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_3_1_2_1"><span class="lab"><font color="Green" title="E77">A5</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E13:101_1_2_1_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">F4</font></span>: 
<font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">cP9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:101_1_2_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E76">E4</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/><a NAME="E14:101_1_2_1_1_2_3_1_2"/>
<font color="Maroon" title="c10">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">cP9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_3_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">xx</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">cP9</font> or <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_3_1_2_2"/><span class="lab"><font color="Green" title="E78">A5</font></span>: 
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">cP9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E2:101_1_2_1_1_2_3_1_2_2"/><span class="lab"><font color="Green" title="E79">B5</font></span>: 
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_3_1_2_2"><span class="lab"><font color="Green" title="E78">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:101_1_2_1_1_2_3_1_2_2"/>
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">cP9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_3_1_2_2"><span class="lab"><font color="Green" title="E78">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:101_1_2_1_1_2_3_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">xx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_1_2_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E73">C3</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E5:101_1_2_1_1_2_3_1_2_2"/><span class="kw">then </span>
not <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:101_1_2_1_1_2_3_1_2_2"/>
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_2_1_1_2_3_1_2_2"><span class="lab"><font color="Green" title="E79">B5</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E15:101_1_2_1_1_2_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">cP9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:101_1_2_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E77">F4</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">A9</font> = <font color="Maroon" title="c10">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c9">P9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T31" target="_self" title="DILWORTH:th.31">SPAChain</a></span>;<br/><a NAME="E17:101_1_2_1_1_2_3_1_2"/>
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c9">P9</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D6" target="_self" title="DILWORTH:def.6">Lwidth</a></span>;<br/><span class="kw">hence </span><a NAME="E18:101_1_2_1_1_2_3_1_2"/>
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c9">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:101_1_2_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E75">D4a</font></span></a>, <a class="txt" href="#E13:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">I</font></span></a>, <a class="txt" href="#E2:101_1_2_1"><span class="lab"><font color="Green" title="E60">B</font></span></a>, <a class="txt" href="#E11:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">Ha</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">Cp9</font> being   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c9">P9</font><span class="kw"> such that </span><br/><a NAME="E16:101_1_2_1_1_2"/><span class="lab"><font color="Green" title="E71">Z</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c11">Cp9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c9">P9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1"><span class="lab"><font color="Green" title="E59">A</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">Cp</font> = <font color="Maroon" title="c11">Cp9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">cP9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E17:101_1_2_1_1_2"/><span class="lab"><font color="Green" title="E72">Zk</font></span>: 
<font color="Maroon" title="c11">Cp9</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">Z</font></span></a></span>;<br/><a NAME="E18:101_1_2_1_1_2"/>
<font color="Maroon" title="c7">cP9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">E</font></span></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/><a NAME="E19:101_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">Zl</font></span>: 
<font color="Maroon" title="c7">cP9</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">Cp</font> = <font color="Maroon" title="c11">Cp9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">cP9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">cP9c</font></span></a>, <a class="ref" href="dilworth.html#T4" target="_self" title="DILWORTH:th.4">SPpart</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Cp</font> implies <font color="Olive">b<sub>1</sub></font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c3">P</font> )</span><br/><span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E74">A2</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Cp</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c3">P</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:101_1_2_1_1_2_4_1"/>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Cp9</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">cP9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_2_1_1_2_4_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Cp9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c3">P</font></span><br/><div class="add"><a NAME="E2:101_1_2_1_1_2_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c9">P9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D11" target="_self" title="DILWORTH:def.11">LCpart</a></span>;<br/><span class="kw">hence </span><a NAME="E3:101_1_2_1_1_2_4_1_1"/>
<font color="Maroon" title="c14">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T28" target="_self" title="DILWORTH:th.28">SPClique</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_2_1_1_2_4_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">cP9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c3">P</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:101_1_2_1_1_2_4_1_2"/>
<font color="Maroon" title="c14">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_2_1"><span class="lab"><font color="Green" title="E61">C</font></span></a>, <a class="txt" href="#E3:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">E</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">Cp</font> = <font color="Maroon" title="c13">Cp</font> as   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c3">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D11" target="_self" title="DILWORTH:def.11">LCpart</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c14">Cp</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c14">Cp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font></span><br/><a NAME="E23:101_1_2_1_1_2"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">cP9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Cp9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:101_1_2_1_1_2_5"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">cP9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Cp9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:101_1_2_1_1_2_5"/><span class="kw">then </span>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">cP9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">cP9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">cP9c</font></span></a>, <a class="ref" href="xboole_1.html#T44" title="XBOOLE_1:th.44">XBOOLE_1:44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:101_1_2_1_1_2_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">Zl</font></span></a>, <a class="ref" href="xboole_1.html#T60" title="XBOOLE_1:th.60">XBOOLE_1:60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E24:101_1_2_1_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c14">Cp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_2_1"><span class="lab"><font color="Green" title="E60">B</font></span></a>, <a class="txt" href="#E14:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">Y</font></span></a>, <a class="txt" href="#E16:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">Z</font></span></a>, <a class="txt" href="#E17:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">Zk</font></span></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E4:101_1_2"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:101_1_2"><span class="lab"><font color="Green" title="E57">P1</font></span></a>, <a class="txt" href="#E3:101_1_2"><span class="lab"><font color="Green" title="E58">P2</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E5:101_1_2"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_2"><span class="lab"><font color="Green" title="E56">S0</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
