
theorem Th40: :: PCAmin:
for R being with_finite_clique# antisymmetric transitive RelStr,
    A being StableSet of R
 st minimals R c= A holds A = minimals R
proof
 let R be with_finite_clique# antisymmetric transitive RelStr,
     A be StableSet of R such that
A1: minimals R c= A;
  A c= minimals R proof
    let x be object;
    assume A2: x in A;
    then A3: R is non empty;
    reconsider x9 = x as Element of R by A2;
     consider y being Element of R such that
  A4: y is_minimal_in [#]R and
  A5: y = x9 or y < x9 by A3,Th36;
  A6: y = x9 or y <= x9 by A5;
      y in minimals R by A3,A4,Def9;
    hence x in minimals R by A1,A2,A6,Def2;
  end;
  hence A = minimals R by A1;
end;
