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 Th4:
  S is calculating_type implies for w1, w2 st w1.1 = w2.1 holds
  GEN(w1, the InitS of S), GEN(w2, the InitS of S) are_c=-comparable
proof
  assume
A1: S is calculating_type;
  let w1, w2 such that
A2: w1.1 = w2.1;
  set A = Seg(1+len w1) /\ Seg(1+len w2);
  1+len w1 <= 1+len w2 or 1+len w2 <= 1+len w1;
  then
A3: Seg(1+len w1) c= Seg(1+len w2) & A = Seg(1+len w1) or
  Seg(1+len w2) c= Seg(1+len w1) & A = Seg(1+len w2) by FINSEQ_1:5,7;
A4: len GEN(w1, the InitS of S) = 1+len w1 by FSM_1:def 2;
A5: len GEN(w2, the InitS of S) = 1+len w2 by FSM_1:def 2;
A6: dom GEN(w1, the InitS of S) = Seg(1+len w1) by A4,FINSEQ_1:def 3;
A7: dom GEN(w2, the InitS of S) = Seg(1+len w2) by A5,FINSEQ_1:def 3;
  now
    let x be object;
    assume
A8: x in A;
    then reconsider i = x as Element of NAT;
A9: i >= 1 by A3,A8,FINSEQ_1:1;
A10: i <= 1+len w1 by A3,A8,FINSEQ_1:1;
    i <= 1+len w2 by A3,A8,FINSEQ_1:1;
    hence GEN(w1, the InitS of S).x = GEN(w2, the InitS of S).x
    by A1,A2,A9,A10;
  end;
  hence GEN(w1, the InitS of S) c= GEN(w2, the InitS of S) or
  GEN(w2, the InitS of S) c= GEN(w1, the InitS of S) by A3,A6,A7,GRFUNC_1:2;
end;
