<?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="firebrick">::  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="firebrick">::  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">Th51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_2"/><span class="lab"><font color="Green" title="E56">A1</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="firebrick">::  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">A2</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">A3</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="firebrick">::  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">A4</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="firebrick">::  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="firebrick">::  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">A5</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="firebrick">::  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">A6</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">A7</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">Th52</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">A8</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="firebrick">::  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">A6</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">Def11</a>, <a class="txt" href="#E1:101_1_2_1_1_1"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="ref" href="eqrel_1.html#T39" title="EQREL_1:th.39">EQREL_1:39</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="firebrick">::  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">A9</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#T30" title="CARD_1:th.30">CARD_1:30</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">A9</font></span></a>, <a class="txt" href="#E4:101_1_2_1"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="txt" href="#E1:101_1_2_1_1_1"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="ref" href="dilworth.html#T20" target="_self" title="DILWORTH:th.20">Th20</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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="firebrick">::  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">A10</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">A11</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/><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">A10</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="c8">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">A12</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="c8">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">A13</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c8">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">A5</font></span></a>, <a class="ref" href="dilworth.html#T44" target="_self" title="DILWORTH:th.44">Th44</a></span>;<br/><a NAME="E9:101_1_2_1_1_2"/><span class="lab"><font color="Green" title="E67">A14</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c8">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">A15</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c8">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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">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="c8">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">A16</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">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="c8">P9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D6" target="_self" title="DILWORTH:def.6">Def6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">A9</font> = <font color="Maroon" title="c9">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">A16</font></span></a>, <a class="ref" href="dilworth.html#T30" target="_self" title="DILWORTH:th.30">Th30</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">A17</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="c9">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="c11">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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="c11">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">A18</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c11">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="c9">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">A12</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="c11">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">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">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="c11">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">A19</font></span>: 
<font color="Maroon" title="c12">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="firebrick">::  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="c11">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="c11">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">A20</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">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">A21</font></span>: 
 not <font color="Maroon" title="c11">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">A12</font></span></a>, <a class="txt" href="#E1:101_1_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E71">A18</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="c12">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="c12">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">A20</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">A21</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">A19</font></span></a>, <a class="txt" href="#E5:101_1_2_1"><span class="lab"><font color="Green" title="E62">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">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">A22</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">S</font>,<font color="Maroon" title="c12">Ca</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D13" target="_self" title="DILWORTH:def.13">Def13</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">S</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c13">S</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c13">S</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">S</font> &amp; <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">S</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">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="c11">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="c13">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">S</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">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="c11">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">A23</font></span>: 
 not <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">A24</font></span>: 
<font color="Maroon" title="c13">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="c13">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">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="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  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="c13">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">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="firebrick">::  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">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">S</font> &amp; <font color="Maroon" title="c14">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">S</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">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="firebrick">::  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="c13">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">Ca</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">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="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  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="c13">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">Ca</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">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">A25</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">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">Ca</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">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">A26</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">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">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">A25</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="c14">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="c14">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">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">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">A26</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">A23</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="firebrick">::  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">A27</font></span>: 
<font color="Maroon" title="c13">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="c13">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">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">A24</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="c14">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="c13">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">A28</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">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">A29</font></span>: 
 not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">S</font>,<font color="Maroon" title="c14">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">A6</font></span></a>, <a class="ref" href="dilworth.html#D13" target="_self" title="DILWORTH:def.13">Def13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">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">A30</font></span>: 
<font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">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">A31</font></span>: 
<font color="Maroon" title="c13">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="c15">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">A32</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="c14">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="c15">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">A29</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="c14">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="c15">c</font> holds <br/><font color="Maroon" title="c12">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="c16">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">p</font> &amp; <font color="Maroon" title="c16">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">c</font> implies <font color="Maroon" title="c12">Ca</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c16">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">A33</font></span>: 
<font color="Maroon" title="c16">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">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">A34</font></span>: 
<font color="Maroon" title="c16">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">Ca</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c16">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="c12">Ca</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c16">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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">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">A35</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c16">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">A36</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">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">A35</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">A37</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">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">A35</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="c17">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="c17">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">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">A36</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="c17">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="firebrick">::  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="c17">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="c16">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">A37</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">A33</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">A34</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">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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="c17">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">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="firebrick">::  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="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="c11">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="c13">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">A37</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">A33</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">A23</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:101_1_2_1_1_2_2_1_2_3"/>
<font color="Maroon" title="c12">Ca</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c16">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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="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">A30</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">A31</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">A28</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">A27</font></span></a>, <a class="txt" href="#E7:101_1_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E72">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:101_1_2_1_1_2_2_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">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="c11">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="c13">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="c11">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">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c11">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">A38</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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">A39</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="c9">A</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c11">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">A17</font></span></a>);<br/></span>
<a NAME="E9:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E73">A40</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">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">A38</font></span></a>, <a class="ref" href="finset_1.html#T8" title="FINSET_1:th.8">FINSET_1:8</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">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="c11">f</font></span>)</span>;<br/>
<a NAME="E10:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E74">A41</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="c11">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="c13">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c11">f</font> implies <font color="Maroon" title="c13">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="c13">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="c11">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">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="c14">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">A42</font></span>: 
<font color="Maroon" title="c14">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="c11">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">A43</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<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="E6:101_1_2_1_1_2_2_2"/><span class="lab"><font color="Green" title="E77">A44</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</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="E7:101_1_2_1_1_2_2_2"/>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> &amp; <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">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">A38</font></span></a>, <a class="txt" href="#E3:101_1_2_1_1_2_2_2"><span class="lab"><font color="Green" title="E75">A42</font></span></a>, <a class="txt" href="#E8:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E72">A39</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="c13">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">A44</font></span></a>, <a class="txt" href="#E4:101_1_2_1_1_2_2_2"><span class="lab"><font color="Green" title="E76">A43</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E75">A45</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="c11">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="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">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="c11">f</font></span>)</span> or <font color="Maroon" title="c13">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="c13">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="c11">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">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="c14">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">A46</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">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">A47</font></span>: 
<font color="Maroon" title="c14">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="c11">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="c15">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">A48</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="c11">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">A49</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="txt" href="#E4:101_1_2_1_1_2_2_3"><span class="lab"><font color="Green" title="E77">A47</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</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="E9:101_1_2_1_1_2_2_3"/><span class="lab"><font color="Green" title="E80">A50</font></span>: 
<font color="Maroon" title="c11">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="E10:101_1_2_1_1_2_2_3"/>
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">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">A38</font></span></a>, <a class="txt" href="#E6:101_1_2_1_1_2_2_3"><span class="lab"><font color="Green" title="E78">A48</font></span></a>, <a class="txt" href="#E8:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E72">A39</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="c13">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">A46</font></span></a>, <a class="txt" href="#E7:101_1_2_1_1_2_2_3"><span class="lab"><font color="Green" title="E79">A49</font></span></a>, <a class="txt" href="#E9:101_1_2_1_1_2_2_3"><span class="lab"><font color="Green" title="E80">A50</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E76">A51</font></span>: 
<font color="Maroon" title="c10">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="c11">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="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A9</font> or <font color="Maroon" title="c13">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="c11">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">A52</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">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="c11">f</font></span>)</span></span><br/>

<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="E3:101_1_2_1_1_2_2_4"/>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">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">A53</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">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="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="c13">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">A39</font></span></a></span>;<br/>
<a NAME="E6:101_1_2_1_1_2_2_4"/>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">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="c11">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">A52</font></span></a>, <a class="txt" href="#E7:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E71">A38</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:101_1_2_1_1_2_2_4"/>
<font color="Maroon" title="c13">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="c11">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">A53</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="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="c11">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">A41</font></span></a>, <a class="txt" href="#E9:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E73">A40</font></span></a>, <a class="txt" href="#E11:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E75">A45</font></span></a>, <a class="ref" href="finset_1.html#T7" title="FINSET_1:th.7">FINSET_1:7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">SSp</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c13">SS</font>;<br/>
<a NAME="E14:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E77">A54</font></span>: 
<font color="Maroon" title="c13">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="c13">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="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="c13">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">A55</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="E17:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E79">A56</font></span>: 
 not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">SS</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">A6</font></span></a>, <a class="ref" href="dilworth.html#D13" target="_self" title="DILWORTH:def.13">Def13</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="E19:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E80">A57</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="E20:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E81">A58</font></span>: 
<font color="Maroon" title="c13">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="c16">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">A59</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="#E17:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E79">A56</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">c</font> = <font color="Maroon" title="c16">c</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c15">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">A57</font></span></a></span>;<br/>
<a NAME="E23:101_1_2_1_1_2_2"/>
<font color="Maroon" title="c10">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A9</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">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">A51</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="c18">A</font> = <font color="Maroon" title="c10">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="c13">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">Th31</a></span>;<br/>
<a NAME="E25:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E83">A60</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="c13">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="c18">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D6" target="_self" title="DILWORTH:def.6">Def6</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="c15">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="c13">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">Th46</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="c15">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="c18">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">A60</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="c19">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c18">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">A61</font></span>: 
<font color="Maroon" title="c17">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c18">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="c19">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">Th48</a>, <a class="txt" href="#E16:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E78">A55</font></span></a>, <a class="txt" href="#E3:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E69">A16</font></span></a>, <a class="txt" href="#E1:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E68">A15</font></span></a>, <a class="txt" href="#E2:101_1_2_1"><span class="lab"><font color="Green" title="E60">A5</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="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c18">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">A61</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">A62</font></span>: 
<font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">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="c20">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">A63</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">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">A64</font></span>: 
<font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">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">A65</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c20">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="c19">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">A39</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="c20">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="c20">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">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="c15">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">A66</font></span>: 
<font color="Maroon" title="c20">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">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="c22">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">S</font> or <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">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">A67</font></span>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">SS</font></span><br/>

<a NAME="E2:101_1_2_1_1_2_2_5"/>
<font color="Maroon" title="c20">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="c11">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">A63</font></span></a>, <a class="txt" href="#E7:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E71">A38</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:101_1_2_1_1_2_2_5"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">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">A67</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="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="c15">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="c20">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">A54</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="c22">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="c20">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="c23">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="firebrick">::  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="c22">Sp</font> implies <font color="Maroon" title="c23">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="c20">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="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Sp</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">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="c20">S</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">e</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c15">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">A68</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">e</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">S</font>
 <span class="kw">and </span><br/><a NAME="E4:101_1_2_1_1_2_2_6"/>
<font color="Maroon" title="c24">e</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c20">S</font>
 ;<br/>
<a NAME="E5:101_1_2_1_1_2_2_6"/>
<font color="Maroon" title="c24">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="c13">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">Def11</a></span>;<br/>
<a NAME="E6:101_1_2_1_1_2_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c24">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">Th28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:101_1_2_1_1_2_2_6"/>
<font color="Maroon" title="c23">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="c20">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">A68</font></span></a>, <a class="ref" href="dilworth.html#T29" target="_self" title="DILWORTH:th.29">Th29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">Sp</font> = <font color="Maroon" title="c22">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="c20">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D11" target="_self" title="DILWORTH:def.11">Def11</a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<a NAME="E40:101_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E90">A69</font></span>: 
<font color="Maroon" title="c23">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="c15">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">A70</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c23">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="c15">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">A69</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c24">cc</font> = <font color="Maroon" title="c17">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">S</font>;<br/>
<a NAME="E42:101_1_2_1_1_2_2"/>
<font color="Maroon" title="c17">c</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c20">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">A62</font></span></a>, <a class="txt" href="#E33:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E86">A63</font></span></a>, <a class="txt" href="#E34:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E87">A64</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">A71</font></span>: 
<font color="Maroon" title="c17">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Sp</font>
 
;<br/>
<a NAME="E44:101_1_2_1_1_2_2"/>
<font color="Maroon" title="c20">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="c19">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="c17">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">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="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">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="c19">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">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">A72</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">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="c19">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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">S</font></span><br/>

<a NAME="E2:101_1_2_1_1_2_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A73</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">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_7"/><span class="lab"><font color="Green" title="E95">A74</font></span>: 
<font color="Maroon" title="c25">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="c19">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">A72</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="c25">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="c25">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="c19">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">A74</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="c25">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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">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="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">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">A73</font></span></a>, <a class="txt" href="#E36:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E89">A66</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="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">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">A73</font></span></a>, <a class="txt" href="#E20:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E81">A58</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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="c25">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="c19">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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">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="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">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">A62</font></span></a>, <a class="txt" href="#E33:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E86">A63</font></span></a>, <a class="txt" href="#E34:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E87">A64</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">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">A75</font></span>: 
<font color="Maroon" title="c25">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">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">A76</font></span>: 
<font color="Maroon" title="c25">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">c</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">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">A77</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="c19">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="c25">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">A71</font></span></a>, <a class="txt" href="#E41:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E91">A70</font></span></a>, <a class="txt" href="#E16:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E78">A55</font></span></a>, <a class="txt" href="#E35:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E88">A65</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">consider </span><font color="Maroon" title="c26">dd</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c15">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">A78</font></span>: 
<font color="Maroon" title="c25">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">dd</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">S</font>
 <span class="kw">and </span><br/><a NAME="E51:101_1_2_1_1_2_2"/>
<font color="Maroon" title="c26">dd</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c20">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">A75</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="c26">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">A78</font></span></a>, <a class="txt" href="#E47:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E94">A76</font></span></a>, <a class="txt" href="#E21:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E82">A59</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E97">A79</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="c25">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="c20">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">A78</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="c25">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="c19">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="c25">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">A77</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="c27">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">A80</font></span>: 
<font color="Maroon" title="c27">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="c19">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="c25">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">A79</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="c27">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="c19">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">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">A80</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="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">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="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">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">A78</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="c17">c</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c26">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">A62</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">A78</font></span></a>, <a class="txt" href="#E47:101_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E94">A76</font></span></a>, <a class="ref" href="eqrel_1.html#D4" title="EQREL_1:def.4">EQREL_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="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="c8">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">A13</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">A81</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c8">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="c9">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">A82</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">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">Def6</a></span>;<br/><a NAME="E14:101_1_2_1_1_2"/><span class="lab"><font color="Green" title="E70">A83</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c8">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="c9">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="c9">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">A84</font></span>: 
<font color="Maroon" title="c9">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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c8">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="c9">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="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> or <font color="Maroon" title="c10">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">A85</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">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"/><span class="kw">then </span>
 not <font color="Maroon" title="c10">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"><span class="lab"><font color="Green" title="E71">A84</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="c10">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">A85</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:101_1_2_1_1_2_3_1_1"/>
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c8">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">A5</font></span></a>, <a class="txt" href="#E9:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A14</font></span></a>, <a class="txt" href="#E13:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A82</font></span></a>, <a class="ref" href="dilworth.html#T45" target="_self" title="DILWORTH:th.45">Th45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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="c9">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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c8">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="c10">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">A86</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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">A87</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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">A86</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">A88</font></span>: 
<font color="Maroon" title="c10">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">A86</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="c11">A9</font> = <font color="Maroon" title="c9">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="c10">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="c10">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="c9">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">A87</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</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">A89</font></span>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c10">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="c10">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="c10">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="c10">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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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="c10">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">A90</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">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="c9">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="c10">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">A89</font></span></a>, <a class="ref" href="card_2.html#T41" title="CARD_2:th.41">CARD_2:41</a></span>;<br/><a NAME="E11:101_1_2_1_1_2_3_1_2"/><span class="lab"><font color="Green" title="E76">A91</font></span>: 
<font color="Maroon" title="c9">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="c10">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="c9">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="c9">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="c10">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="c12">xx</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c12">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">A92</font></span>: 
<font color="Maroon" title="c12">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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="c10">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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">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="kw">then </span><span class="lab"><font color="Green" title="E78">A93</font></span>: 
<font color="Maroon" title="c12">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 
<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="E3:101_1_2_1_1_2_3_1_2_1"/>
 not <font color="Maroon" title="c12">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="c10">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">A92</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"/><span class="kw">then </span>
<font color="Maroon" title="c12">xx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">x</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="E5:101_1_2_1_1_2_3_1_2_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c12">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">A87</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">A88</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">A93</font></span></a>, <a class="txt" href="#E4:101_1_2_1"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="ref" href="dilworth.html#T15" target="_self" title="DILWORTH:th.15">Th15</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="c12">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">A92</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="firebrick">::  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">A94</font></span>: 
<font color="Maroon" title="c9">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="c10">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="c9">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">A91</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="c9">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="c9">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="c10">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="c12">xx</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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="c12">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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="c10">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">A95</font></span>: 
<font color="Maroon" title="c12">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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="c10">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="kw">then </span><span class="lab"><font color="Green" title="E79">A96</font></span>: 
<font color="Maroon" title="c12">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</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_3_1_2_2"/>
<font color="Maroon" title="c12">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">A95</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="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">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">A88</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="c12">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="c10">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="c12">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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="c10">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">A96</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="firebrick">::  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="c9">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="c10">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="c9">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">A94</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="c12">A9</font> = <font color="Maroon" title="c9">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="c10">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="c8">P9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T31" target="_self" title="DILWORTH:th.31">Th31</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="c8">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="c12">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D6" target="_self" title="DILWORTH:def.6">Def6</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="c8">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">A90</font></span></a>, <a class="txt" href="#E13:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A82</font></span></a>, <a class="txt" href="#E2:101_1_2_1"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="txt" href="#E11:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A81</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">Cp9</font> being   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c8">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">A97</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c10">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="c8">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">A4</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">Cp</font> = <font color="Maroon" title="c10">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">A98</font></span>: 
<font color="Maroon" title="c10">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">A97</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">A11</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">A99</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="c12">Cp</font> = <font color="Maroon" title="c10">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">A12</font></span></a>, <a class="ref" href="dilworth.html#T4" target="_self" title="DILWORTH:th.4">Th4</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="c13">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">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">A100</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cp</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Cp9</font> or <font color="Maroon" title="c13">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">A100</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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Cp9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c13">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c8">P9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D11" target="_self" title="DILWORTH:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E3:101_1_2_1_1_2_4_1_1"/>
<font color="Maroon" title="c13">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">Th28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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="c13">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="firebrick">::  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="c13">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">A6</font></span></a>, <a class="txt" href="#E3:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A11</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">Cp</font> = <font color="Maroon" title="c12">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">Def11</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c13">Cp</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c13">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="c10">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="c10">Cp9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">A12</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">A99</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E24:101_1_2_1_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c13">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">A5</font></span></a>, <a class="txt" href="#E14:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A83</font></span></a>, <a class="txt" href="#E16:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A97</font></span></a>, <a class="txt" href="#E17:101_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A98</font></span></a>, <a class="ref" href="card_2.html#T41" title="CARD_2:th.41">CARD_2:41</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><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">A2</font></span></a>, <a class="txt" href="#E3:101_1_2"><span class="lab"><font color="Green" title="E58">A3</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">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
