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 for P being RedSequence of ==>.-relation(TS1)
  holds P is RedSequence of ==>.-relation(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 & the Tran of TS1 = the Tran
  of TS2;
  let P be RedSequence of ==>.-relation(TS1);
A2: now
    let i be Nat;
    assume i in dom P & i + 1 in dom P;
    then [P.i, P.(i + 1)] in ==>.-relation(TS1) by REWRITE1:def 2;
    hence [P.i, P.(i + 1)] in ==>.-relation(TS2) by A1,Th34;
  end;
  len P > 0;
  hence thesis by A2,REWRITE1:def 2;
end;
