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 Th43:
  for B for p, q, r, s st p^q = r^s & p in B & r in B
      holds p = r & q = s
proof
  let B, p, q, r, s;
  assume that A2: p^q = r^s and A3: p in B & r in B;
  consider t such that A4: p^t = r or p = r^t by Th1, A2;
  thus p = r by A3, A4, Th40;
  hence q = s by A2, FINSEQ_1:33;
end;
