
theorem Th21: :: StaCliCompl:
for R being RelStr, C being StableSet of R holds C is Clique of ComplRelStr R
proof
 let R be RelStr, C be StableSet of R;
   set CR = ComplRelStr R;
A1: C is Subset of CR by NECKLACE:def 8;
  now
    let a, b be Element of CR such that
  A2: a in C and
  A3: b in C and
  A4: a <> b;
     reconsider ap1 = a, bp1 = b as Element of R by NECKLACE:def 8;
     not ap1 <= bp1 & not bp1 <= ap1 by A2,A3,A4,DILWORTH:def 2;
   hence a <= b or b <= a by A2,A4,Th18;
  end;
 hence C is Clique of ComplRelStr R by A1,DILWORTH:6;
end;
