reserve k,m,n for Nat,
  a, b, c for object,
  x, y, X, Y, Z for set,
  D for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R, P1, P2, Q1, Q2, R1, R2 for FinSequence-membered set;
reserve S, T for non empty FinSequence-membered set;
reserve A for Function of P, NAT;
reserve U, V, W for Subset of P*;
reserve k,l,m,n,i,j for Nat,
  a, b, c for object,
  x, y, z, X, Y, Z for set,
  D, D1, D2 for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R for FinSequence-membered set;

theorem Th40:
  for P holds P is antichain-like
      iff for p,q st p in P & p^q in P holds p = p^q
proof
  let P;
  thus P is antichain-like implies for p,q st p in P & p^q in P holds p = p^q
    proof
    assume that A1: P is antichain-like;
    let p, q;
    assume p in P & p^q in P;
    then q = {} by A1;
    hence thesis by FINSEQ_1:34;
    end;
  thus thesis by FINSEQ_1:87;
end;
