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 Th34:
  for TS1 being non empty transition-system over F1, TS2 being non
  empty transition-system over F2 st
the Tran of TS1 = the Tran of TS2 holds ==>.-relation(TS1) = ==>.-relation(TS2)
proof
  let TS1 be non empty transition-system over F1, TS2 be non empty
  transition-system over F2 such that
A1: the Tran of TS1 = the Tran of TS2;
A2: now
    let x be object;
    assume
A3: x in ==>.-relation(TS1);
    then consider s, t being Element of TS1, v, w such that
A4: x = [[s, v], [t, w]] by Th33;
    s, v ==>. t, w, TS1 by A3,A4,Def4;
    then s, v ==>. t, w, TS2 by A1,Th21;
    hence x in ==>.-relation(TS2) by A4,Def4;
  end;
  now
    let x be object;
    assume
A5: x in ==>.-relation(TS2);
    then consider s, t being Element of TS2, v, w such that
A6: x = [[s, v], [t, w]] by Th33;
    s, v ==>. t, w, TS2 by A5,A6,Def4;
    then s, v ==>. t, w, TS1 by A1,Th21;
    hence x in ==>.-relation(TS1) by A6,Def4;
  end;
  hence thesis by A2,TARSKI:2;
end;
