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<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  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</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="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st  <a href="group_1.html#K7" title="GROUP_1:func.7">card</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#K5" title="CARD_1:func.5">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="E1:96"/><span class="lab"><font color="Green" title="E54">P</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_1"/><span class="lab"><font color="Green" title="E55">A1</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="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">P</font> be   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</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="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> 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="c2">P</font> st  <a href="card_1.html#K5" title="CARD_1:func.5">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="c2">P</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:96_1"/><span class="lab"><font color="Green" title="E56">B1</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c2">P</font> st  <a href="card_1.html#K5" title="CARD_1:func.5">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="c2">P</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">wP</font> =  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c2">P</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_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="E1:96_1_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_1"/><span class="lab"><font color="Green" title="E57">S1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c2">P</font> st  <a href="card_1.html#K5" title="CARD_1:func.5">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="c2">P</font></span><br/><div class="add"><a NAME="E2:96_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_1"><span class="lab"><font color="Green" title="E56">B1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">C</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c2">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T54" title="EQREL_1:th.54">EQREL_1:54</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c4">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">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="c2">P</font></span><br/><a NAME="E4:96_1_1_1"/>
<font color="Maroon" title="c2">P</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_1"><span class="lab"><font color="Green" title="E56">B1</font></span></a>, <a class="txt" href="#E1:96_1_1_1"><span class="lab"><font color="Green" title="E57">S1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:96_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_1"><span class="lab"><font color="Green" title="E56">B1</font></span></a>, <a class="txt" href="#E1:96_1_1_1"><span class="lab"><font color="Green" title="E57">S1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2"/><span class="lab"><font color="Green" title="E57">S1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c2">P</font> st  <a href="card_1.html#K5" title="CARD_1:func.5">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="c2">P</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_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="E1:96_1_1_2_1"/>
(  ex <font color="Olive" title="b1">A</font> being  <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c2">P</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">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="c2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="dilworth.html#K6" title="DILWORTH:func.6">maximals</a> <font color="Maroon" title="c2">P</font> ) or for <font color="Olive" title="b1">A</font> being  <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c2">P</font>  holds <br/>(  not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">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="c2">P</font> or <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c2">P</font> or <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K6" title="DILWORTH:func.6">maximals</a> <font color="Maroon" title="c2">P</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2_1_1"/>
 ex <font color="Olive" title="b1">A</font> being  <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c2">P</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">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="c2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="dilworth.html#K6" title="DILWORTH:func.6">maximals</a> <font color="Maroon" title="c2">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c2">P</font> st  <a href="card_1.html#K5" title="CARD_1:func.5">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="c2">P</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">A</font> being   <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c2">P</font><span class="kw"> such that </span><br/><a NAME="E3:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E58">S2a</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">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="c2">P</font>
 <span class="kw">and </span><br/><a NAME="E4:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E59">S2b</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">and </span><br/><a NAME="E5:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E60">S2c</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="dilworth.html#K6" title="DILWORTH:func.6">maximals</a> <font color="Maroon" title="c2">P</font>
 ;<br/><span class="kw">set </span><font color="Maroon" title="c5">aA</font> =  <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c4">A</font>;<br/><span class="kw">set </span><font color="Maroon" title="c6">bA</font> =  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c4">A</font>;<br/><span class="kw">set </span><font color="Maroon" title="c7">cP</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">P</font>;<br/><span class="kw">set </span><font color="Maroon" title="c8">Pa</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c4">A</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">Pb</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c4">A</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">PP</font> = <font color="Maroon" title="c2">P</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</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="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_2"><span class="lab"><font color="Green" title="E57">S1</font></span></a>, <a class="txt" href="#E2:96_1"><span class="lab"><font color="Green" title="E56">B1</font></span></a></span>;<br/><a NAME="E7:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E61">S2d</font></span>: 
 <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c4">A</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> <span class="p2">(<span class="default"><a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c4">A</font></span>)</span></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/><a NAME="E8:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E62">S2e</font></span>: 
 <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c4">A</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> <span class="p2">(<span class="default"><a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c4">A</font></span>)</span></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/><a NAME="E9:96_1_1_2_1_1"/>
not  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c10">PP</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E59">S2b</font></span></a>, <a class="ref" href="dilworth.html#T40" target="_self" title="DILWORTH:th.40">PCAmin</a>, <a class="ref" href="dilworth.html#T26" target="_self" title="DILWORTH:th.26">PCAamin</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">mi</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E63">A2a</font></span>: 
<font color="Maroon" title="c11">mi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">and </span><br/><a NAME="E12:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E64">A2b</font></span>: 
not <font color="Maroon" title="c11">mi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E13:96_1_1_2_1_1"/>
not  <a href="dilworth.html#K6" title="DILWORTH:func.6">maximals</a> <font color="Maroon" title="c10">PP</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">S2c</font></span></a>, <a class="ref" href="dilworth.html#T41" target="_self" title="DILWORTH:th.41">PCAmax</a>, <a class="ref" href="dilworth.html#T27" target="_self" title="DILWORTH:th.27">PCAbmax</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">ma</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E15:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E65">B2a</font></span>: 
<font color="Maroon" title="c12">ma</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K6" title="DILWORTH:func.6">maximals</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">and </span><br/><a NAME="E16:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E66">B2b</font></span>: 
not <font color="Maroon" title="c12">ma</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">Pa</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c4">A</font></span>)</span> as   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</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="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><a NAME="E18:96_1_1_2_1_1"/>
<font color="Maroon" title="c11">mi</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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A2a</font></span></a></span>;<br/><a NAME="E19:96_1_1_2_1_1"/><span class="kw">then </span>
 <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c4">A</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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E64">A2b</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><a NAME="E20:96_1_1_2_1_1"/><span class="kw">then </span>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c13">Pa</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E61">S2d</font></span></a>, <a class="ref" href="card_2.html#T67" title="CARD_2:th.67">CARD_2:67</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">Ca</font> being   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c13">Pa</font><span class="kw"> such that </span><br/><a NAME="E22:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E67">C2a</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">Ca</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="c13">Pa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E2:96_1"><span class="lab"><font color="Green" title="E56">B1</font></span></a></span>;<br/><a NAME="E23:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E68">C2b</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c13">Pa</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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E58">S2a</font></span></a>, <a class="ref" href="dilworth.html#T45" target="_self" title="DILWORTH:th.45">Wsubp1</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">Pb</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c4">A</font></span>)</span> as   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</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="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><a NAME="E25:96_1_1_2_1_1"/>
<font color="Maroon" title="c12">ma</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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">B2a</font></span></a></span>;<br/><a NAME="E26:96_1_1_2_1_1"/><span class="kw">then </span>
 <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c4">A</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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">B2b</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><a NAME="E27:96_1_1_2_1_1"/><span class="kw">then </span>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c15">Pb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E62">S2e</font></span></a>, <a class="ref" href="card_2.html#T67" title="CARD_2:th.67">CARD_2:67</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">L</font> being   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c15">Pb</font><span class="kw"> such that </span><br/><a NAME="E29:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E69">D2a</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">L</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="c15">Pb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E2:96_1"><span class="lab"><font color="Green" title="E56">B1</font></span></a></span>;<br/><a NAME="E30:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E70">D2b</font></span>: 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c15">Pb</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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E58">S2a</font></span></a>, <a class="ref" href="dilworth.html#T45" target="_self" title="DILWORTH:th.45">Wsubp1</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">C</font> being   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c2">P</font><span class="kw"> such that </span><br/><a NAME="E32:96_1_1_2_1_1"/><span class="lab"><font color="Green" title="E71">Z</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">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="c10">PP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E58">S2a</font></span></a>, <a class="txt" href="#E22:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">C2a</font></span></a>, <a class="txt" href="#E23:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">C2b</font></span></a>, <a class="txt" href="#E29:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E69">D2a</font></span></a>, <a class="txt" href="#E30:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E70">D2b</font></span></a>, <a class="ref" href="dilworth.html#T49" target="_self" title="DILWORTH:th.49">Main</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c17">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">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="c2">P</font></span><br/><span class="kw">thus </span><a NAME="E33:96_1_1_2_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:96_1_1_2_1_1"><span class="lab"><font color="Green" title="E71">Z</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E58">S2</font></span>: 
for <font color="Olive" title="b1">A</font> being  <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c2">P</font>  holds <br/>(  not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">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="c2">P</font> or <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c2">P</font> or <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K6" title="DILWORTH:func.6">maximals</a> <font color="Maroon" title="c2">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c2">P</font> st  <a href="card_1.html#K5" title="CARD_1:func.5">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="c2">P</font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c4">S</font> being   <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c2">P</font><span class="kw"> such that </span><br/><a NAME="E3:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E59">B2</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">S</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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D6" target="_self" title="DILWORTH:def.6">Lwidth</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">PP</font> = <font color="Maroon" title="c2">P</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</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="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_2"><span class="lab"><font color="Green" title="E57">S1</font></span></a>, <a class="txt" href="#E2:96_1"><span class="lab"><font color="Green" title="E56">B1</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">iR</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c5">PP</font>;<br/><span class="kw">set </span><font color="Maroon" title="c7">cR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">PP</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c2">P</font>;<br/><a NAME="E6:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E60">D2a</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c5">PP</font>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">PP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E60">D2a</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">PP</font><span class="kw"> such that </span><br/><a NAME="E9:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E61">F2</font></span>: 
<font color="Maroon" title="c10">b</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">PP</font>
 <span class="kw">and </span><br/><a NAME="E10:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E62">G2</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> or <font color="Maroon" title="c9">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T38" target="_self" title="DILWORTH:th.38">Pminmax</a></span>;<br/><a NAME="E11:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E63">F2a</font></span>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K6" title="DILWORTH:func.6">maximals</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E61">F2</font></span></a>, <a class="ref" href="dilworth.html#D10" target="_self" title="DILWORTH:def.10">Lmax</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">cP</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">P</font>;<br/><span class="kw">set </span><font color="Maroon" title="c12">cPP</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">PP</font>;<br/><span class="kw">set </span><font color="Maroon" title="c13">Cn</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">cP9</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>;<br/><a NAME="E12:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E64">G2a</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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="E13:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E65">G2b</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/><a NAME="E14:96_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">G2c</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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="c2">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E64">G2a</font></span></a>, <a class="ref" href="xboole_1.html#T88" title="XBOOLE_1:th.88">XBOOLE_1:88</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">cP9</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">PP</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c16">P9</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c15">cP9</font>;<br/><a NAME="E16:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E67">H2a</font></span>: 
<font color="Maroon" title="c15">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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c15">cP9</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">set </span><font color="Maroon" title="c17">k</font> =  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c15">cP9</font></span>)</span>;<br/><a NAME="E17:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E68">H2</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cP9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c15">cP9</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/><a NAME="E18:96_1_1_2_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cP9</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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">C</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="c15">cP9</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E20:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E69">I2</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c15">cP9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E2:96_1"><span class="lab"><font color="Green" title="E56">B1</font></span></a>, <a class="txt" href="#E17:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E68">H2</font></span></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>;<br/><a NAME="E21:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E70">I2a</font></span>: 
not <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:96_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E71">A3</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:96_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E72">B3</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:96_1_1_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E65">G2b</font></span></a>, <a class="ref" href="zfmisc_1.html#T55" title="ZFMISC_1:th.55">ZFMISC_1:55</a>, <a class="txt" href="#E16:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E67">H2a</font></span></a>, <a class="txt" href="#E1:96_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E2:96_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E72">B3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c19">wP</font> =  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c5">PP</font>;<br/><span class="kw">set </span><font color="Maroon" title="c20">wP1</font> = <span class="p1">(<span class="default"><a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c5">PP</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/><a NAME="E22:96_1_1_2_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <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="c5">PP</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/><a NAME="E23:96_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c5">PP</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E24:96_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c5">PP</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">wP1</font> = <span class="p1">(<span class="default"><a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c5">PP</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c22">S9</font> = <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">cP9</font>;<br/><a NAME="E26:96_1_1_2_1_2"/>
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</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/><a NAME="E27:96_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">J2aa</font></span>: 
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">cP9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></span>;<br/><a NAME="E28:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E72">J2ab</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <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="c9">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">b</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="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">S9</font> = <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">cP9</font> as   <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="c15">cP9</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T31" target="_self" title="DILWORTH:th.31">SPAChain</a></span>;<br/><a NAME="E30:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E73">J2a</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c23">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">wP1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_2_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="E1:96_1_1_2_1_2_2_1"/>
( <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c2">P</font> or <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K6" title="DILWORTH:func.6">maximals</a> <font color="Maroon" title="c5">PP</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E58">S2</font></span></a>, <a class="txt" href="#E3:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E59">B2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2_1_2_2_1_1"/><span class="lab"><font color="Green" title="E74">S3</font></span>: 
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c2">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c23">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">wP1</font></span><br/><div class="add"><a NAME="E2:96_1_1_2_1_2_2_1_1"/>
<font color="Maroon" title="c23">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_1_2_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_2_1_1_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="E1:96_1_1_2_1_2_2_1_1_1_1"/>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> or <font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2_1_2_2_1_1_1_1_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:96_1_1_2_1_2_2_1_1_1_1_1"/>
<font color="Maroon" title="c23">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">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="#E27:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E71">J2aa</font></span></a>, <a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2_1_2_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E75">A5aa</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_1_2_2_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:96_1_1_2_1_2_2_1_1_1_1_2_1"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c5">PP</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:96_1_1_2_1_2_2_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">b</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">PP</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D9" target="_self" title="DILWORTH:def.9">Lmin</a></span>;<br/><span class="kw">hence </span><a NAME="E3:96_1_1_2_1_2_2_1_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_2_1_2_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E75">A5aa</font></span></a>, <a class="txt" href="#E10:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E62">G2</font></span></a>, <a class="ref" href="waybel_4.html#T57" title="WAYBEL_4:th.57">WAYBEL_4:57</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:96_1_1_2_1_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c23">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">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="#E27:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E71">J2aa</font></span></a>, <a class="txt" href="#E28:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E72">J2ab</font></span></a>, <a class="txt" href="#E1:96_1_1_2_1_2_2_1_1"><span class="lab"><font color="Green" title="E74">S3</font></span></a>, <a class="ref" href="dilworth.html#T1" target="_self" title="DILWORTH:th.1">set00</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c23">S9</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_2_1_2_2_1_1"><span class="lab"><font color="Green" title="E74">S3</font></span></a>, <a class="ref" href="euler_1.html#T5" title="EULER_1:th.5">EULER_1:5</a></span>
<br/>.= 
<font color="Maroon" title="c21">wP1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E59">B2</font></span></a>, <a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2_1_2_2_1_2"/><span class="lab"><font color="Green" title="E74">S3</font></span>: 
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K6" title="DILWORTH:func.6">maximals</a> <font color="Maroon" title="c5">PP</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c23">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">wP1</font></span><br/><div class="add"><a NAME="E2:96_1_1_2_1_2_2_1_2"/><span class="lab"><font color="Green" title="E75">A4</font></span>: 
<font color="Maroon" title="c23">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_1_2_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_2_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="E1:96_1_1_2_1_2_2_1_2_1_1"/>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> or <font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2_1_2_2_1_2_1_1_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:96_1_1_2_1_2_2_1_2_1_1_1"/>
<font color="Maroon" title="c23">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">b</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="#E27:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E71">J2aa</font></span></a>, <a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2_1_2_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E76">A5aa</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_1_2_2_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:96_1_1_2_1_2_2_1_2_1_1_2_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K6" title="DILWORTH:func.6">maximals</a> <font color="Maroon" title="c5">PP</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:96_1_1_2_1_2_2_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">a</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">PP</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D10" target="_self" title="DILWORTH:def.10">Lmax</a></span>;<br/><span class="kw">hence </span><a NAME="E3:96_1_1_2_1_2_2_1_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_2_1_2_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A5aa</font></span></a>, <a class="txt" href="#E10:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E62">G2</font></span></a>, <a class="ref" href="waybel_4.html#T56" title="WAYBEL_4:th.56">WAYBEL_4:56</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:96_1_1_2_1_2_2_1_2_1_1_2"/>
<font color="Maroon" title="c23">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">b</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="#E27:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E71">J2aa</font></span></a>, <a class="txt" href="#E28:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E72">J2ab</font></span></a>, <a class="txt" href="#E1:96_1_1_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E74">S3</font></span></a>, <a class="ref" href="dilworth.html#T1" target="_self" title="DILWORTH:th.1">set00</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E3:96_1_1_2_1_2_2_1_2"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E74">S3</font></span></a>, <a class="txt" href="#E9:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E61">F2</font></span></a>, <a class="ref" href="dilworth.html#D10" target="_self" title="DILWORTH:def.10">Lmax</a></span>;<br/><span class="kw">hence </span> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c23">S9</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_1_1_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E75">A4</font></span></a>, <a class="ref" href="euler_1.html#T5" title="EULER_1:th.5">EULER_1:5</a></span>
<br/>.= 
<font color="Maroon" title="c21">wP1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E59">B2</font></span></a>, <a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E31:96_1_1_2_1_2"/><span class="lab"><font color="Green" title="E74">J2b</font></span>: 
for <font color="Olive" title="b1">T</font> being  <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="c15">cP9</font></span>)</span> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">T</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="c23">S9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">T</font> be   <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="c15">cP9</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c24">T</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="c23">S9</font></span><br/>

<span class="kw">assume </span><a NAME="E1:96_1_1_2_1_2_3"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c24">T</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c23">S9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:96_1_1_2_1_2_3"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c24">T</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c23">S9</font></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="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">V</font> being   <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="c15">cP9</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:96_1_1_2_1_2_3"/><span class="lab"><font color="Green" title="E75">B3</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">V</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> <font color="Maroon" title="c23">S9</font></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="ref" href="dilworth.html#T17" target="_self" title="DILWORTH:th.17">AChain1</a></span>;<br/>
<a NAME="E5:96_1_1_2_1_2_3"/><span class="lab"><font color="Green" title="E76">C3</font></span>: 
<font color="Maroon" title="c25">V</font> is   <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T30" target="_self" title="DILWORTH:th.30">SPAChain1</a></span>;<br/>
<a NAME="E6:96_1_1_2_1_2_3"/>
( <font color="Maroon" title="c25">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c2">P</font> or <font color="Maroon" title="c25">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K6" title="DILWORTH:func.6">maximals</a> <font color="Maroon" title="c2">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E58">S2</font></span></a>, <a class="txt" href="#E5:96_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E76">C3</font></span></a>, <a class="txt" href="#E4:96_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E75">B3</font></span></a>, <a class="txt" href="#E30:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E73">J2a</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:96_1_1_2_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E60">D2a</font></span></a>, <a class="txt" href="#E11:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E63">F2a</font></span></a>, <a class="txt" href="#E16:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E67">H2a</font></span></a>, <a class="txt" href="#E13:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E65">G2b</font></span></a>, <a class="ref" href="zfmisc_1.html#T55" title="ZFMISC_1:th.55">ZFMISC_1:55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="lab"><font color="Green" title="E75">J2c</font></span>: <span class="p1">(<span class="default"><a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c15">cP9</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c5">PP</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</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="#E30:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E73">J2a</font></span></a>, <a class="txt" href="#E31:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E74">J2b</font></span></a>, <a class="ref" href="dilworth.html#D6" target="_self" title="DILWORTH:def.6">Lwidth</a></span>
<br/>.= 
 <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c5">PP</font>

;<br/><span class="kw">set </span><font color="Maroon" title="c24">CC</font> = <font color="Maroon" title="c18">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"><span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E33:96_1_1_2_1_2"/>
<font color="Maroon" title="c15">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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E66">G2c</font></span></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/><a NAME="E34:96_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">x</font></span>: 
<font color="Maroon" title="c15">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="c2">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_1_2_5"><span class="kw">now </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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">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"><span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> 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="c2">P</font> )</span><br/><span class="kw">assume </span><a NAME="E1:96_1_1_2_1_2_5"/><span class="lab"><font color="Green" title="E77">A3</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">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"><span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c2">P</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_5_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="E1:96_1_1_2_1_2_5_1"/>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">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"><span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></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="#E1:96_1_1_2_1_2_5"><span class="lab"><font color="Green" title="E77">A3</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="96_1_1_2_1_2_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2_1_2_5_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c2">P</font></span><br/><div class="add"><a NAME="E2:96_1_1_2_1_2_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c25">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="c15">cP9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D11" target="_self" title="DILWORTH:def.11">LCpart</a></span>;<br/><span class="kw">hence </span><a NAME="E3:96_1_1_2_1_2_5_1_1"/>
<font color="Maroon" title="c25">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T28" target="_self" title="DILWORTH:th.28">SPClique</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2_1_2_5_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"><span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c2">P</font></span><br/><div class="add"><a NAME="E2:96_1_1_2_1_2_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A4</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_5_1_2_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="E1:96_1_1_2_1_2_5_1_2_1"/>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> or <font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_5_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2_1_2_5_1_2_1_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c2">P</font></span><br/><div class="add"><a NAME="E2:96_1_1_2_1_2_5_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c25">x</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="c9">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="#E2:96_1_1_2_1_2_5_1_2"><span class="lab"><font color="Green" title="E78">A4</font></span></a>, <a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></span>;<br/><span class="kw">hence </span><a NAME="E3:96_1_1_2_1_2_5_1_2_1_1"/>
<font color="Maroon" title="c25">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c2">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2_1_2_5_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2_1_2_5_1_2_1_2"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c2">P</font></span><br/><div class="add"><a NAME="E2:96_1_1_2_1_2_5_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E62">G2</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:96_1_1_2_1_2_5_1_2_1_2"/>
<font color="Maroon" title="c25">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_1_1_2_1_2_5_1_2"><span class="lab"><font color="Green" title="E78">A4</font></span></a>, <a class="ref" href="dilworth.html#T8" target="_self" title="DILWORTH:th.8">Clique36b</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></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="c25">CC</font> = <font color="Maroon" title="c18">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"><span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></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="c2">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E66">G2c</font></span></a>, <a class="txt" href="#E16:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E67">H2a</font></span></a>, <a class="ref" href="dilworth.html#T4" target="_self" title="DILWORTH:th.4">SPpart</a>, <a class="ref" href="dilworth.html#D11" target="_self" title="DILWORTH:def.11">LCpart</a>, <a class="txt" href="#E34:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E76">x</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c25">CC</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">CC</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="c2">P</font></span><br/><span class="kw">thus </span><a NAME="E37:96_1_1_2_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">CC</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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E69">I2</font></span></a>, <a class="txt" href="#E32:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E75">J2c</font></span></a>, <a class="txt" href="#E21:96_1_1_2_1_2"><span class="lab"><font color="Green" title="E70">I2a</font></span></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:96"/><span class="lab"><font color="Green" title="E55">Z</font></span>: 
for <font color="Olive" title="b1">n</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">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E54">P</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</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="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> st  <a href="card_1.html#K5" title="CARD_1:func.5">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/>

<a NAME="E3:96"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">R</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:96"/>
 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#K5" title="CARD_1:func.5">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="#E2:96"><span class="lab"><font color="Green" title="E55">Z</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
