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;
