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;
reserve B, C for antichain;
reserve S, T for Polish-language;

theorem Th44:
  for P st for p, q st p in P & q in P holds dom p = dom q
      holds P is antichain-like
proof
  let P;
  assume that A1: for p, q st p in P & q in P holds dom p = dom q;
  for p, q st p in P & p^q in P holds p = p^q
    proof
    let p, q;
    assume that A2: p in P & p^q in P;
    set r = p^q;
    dom p = dom r by A1, A2;
    then p = r | (dom r) by FINSEQ_1:21 .= r;
    hence thesis;
    end;
  hence thesis by Th40;
end;
