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 Th2:
  GEN(<*s1,s2*>, q) = <*q, (the Tran of S).[q, s1],
  (the Tran of S).[(the Tran of S).[q, s1], s2]*>
proof
  set w = <*s1,s2*>;
A1: GEN(w,q).1 = q by FSM_1:def 2;
A2: w = <*s1*>^<*s2*> by FINSEQ_1:def 9;
A3: len <*s1*> = 1 by FINSEQ_1:39;
  GEN(<*s1*>, q) = <*q, (the Tran of S).[q, s1]*> by Th1;
  then GEN(<*s1*>, q).(1+1) = (the Tran of S).[q, s1];
  then q, <*s1*>-leads_to ((the Tran of S).[q, s1]) by A3;
  then
A4: GEN(w, q).(1+1) = (the Tran of S).[q, s1] by A2,A3,FSM_1:6;
A5: len w = 2 by FINSEQ_1:44;
  2 <= len w by FINSEQ_1:44;
  then consider WI being Element of I, QI, QI1 being State of S such that
A6: WI = w.2 and
A7: QI = GEN(w, q).2 and
A8: QI1 = GEN(w, q).(2+1) and
A9: WI-succ_of QI = QI1 by FSM_1:def 2;
A10: WI = s2 by A6;
  len GEN(w, q) = len w + 1 by FSM_1:def 2
    .= 3 by A5;
  hence thesis by A1,A4,A7,A8,A9,A10,FINSEQ_1:45;
end;
