reserve x, x1, x2, y, y1, y2, z, z1, z2 for object, X, X1, X2 for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u9, u1, u2, v, v1, v2, w, w1, w2 for Element of E^omega;
reserve F, F1, F2 for Subset of E^omega;
reserve i, k, l, n for Nat;
reserve TS for non empty transition-system over F;
reserve s, s9, s1, s2, t, t1, t2 for Element of TS;
reserve S for Subset of TS;

theorem
  for TS1 being non empty transition-system over F1, TS2 being non empty
transition-system over F2 st the carrier of TS1 = the carrier of TS2 & the Tran
  of TS1 = the Tran of TS2 holds x-succ_of (X, TS1) = x-succ_of (X, TS2)
proof
  let TS1 be non empty transition-system over F1, TS2 be non empty
  transition-system over F2 such that
A1: the carrier of TS1 = the carrier of TS2 and
A2: the Tran of TS1 = the Tran of TS2;
A3: now
    let y be object;
    assume
A4: y in x-succ_of (X, TS2);
    then reconsider q = y as Element of TS2;
    consider p being Element of TS2 such that
A5: p in X and
A6: p, x ==>* q, TS2 by A4,Th103;
    reconsider q as Element of TS1 by A1;
    reconsider p as Element of TS1 by A1;
    p, x ==>* q, TS1 by A1,A2,A6,Th94;
    hence y in x-succ_of (X, TS1) by A5;
  end;
  now
    let y be object;
    assume
A7: y in x-succ_of (X, TS1);
    then reconsider q = y as Element of TS1;
    consider p being Element of TS1 such that
A8: p in X and
A9: p, x ==>* q, TS1 by A7,Th103;
    reconsider q as Element of TS2 by A1;
    reconsider p as Element of TS2 by A1;
    p, x ==>* q, TS2 by A1,A2,A9,Th94;
    hence y in x-succ_of (X, TS2) by A8;
  end;
  hence thesis by A3,TARSKI:2;
end;
