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 Th45:
  for P, a st for p st p in P holds dom p = a holds P is antichain-like
proof
  let P, a;
  assume that A1: for p st p in P holds dom p = a;
  for p, q st p in P & q in P holds dom p = dom q
  proof
    let p, q;
    assume that A2: p in P and A3: q in P;
    dom p = a by A1, A2 .= dom q by A1, A3;
    hence thesis;
  end;
  hence thesis by Th44;
end;
