reserve x,y for Real,
  i, j for non zero Element of NAT,
  I, O for non empty set,
  s,s1,s2,s3 for Element of I,
  w, w1, w2 for FinSequence of I,
  t for Element of O,
  S for non empty FSM over I,
  q, q1 for State of S;

theorem
  S is calculating_type implies
  for w1, w2 st w1.1 = w2.1 & len w1 = len w2 holds
  GEN(w1, the InitS of S) = GEN(w2, the InitS of S)
proof
  assume
A1: S is calculating_type;
  let w1,w2;
  assume that
A2: w1.1 = w2.1 and
A3: len w1 = len w2;
A4: len GEN(w1, the InitS of S) = 1 + len w1 by FSM_1:def 2;
  len GEN(w2, the InitS of S) = 1 + len w2 by FSM_1:def 2;
  hence thesis by A1,A2,A3,A4,Th4,TREES_1:4;
end;
