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 Th1:
  GEN(<*s*>, q) = <*q, (the Tran of S).[q, s]*>
proof
A1: len <*s*> = 1 by FINSEQ_1:39;
A2: len GEN(<*s*>, q) = len <*s*> + 1 by FSM_1:def 2
    .= 2 by A1;
A3: GEN(<*s*>, q).1 = q by FSM_1:def 2;
  1 <= len <*s*> by FINSEQ_1:39;
  then consider WI being Element of I, QI, QI1 being State of S such that
A4: WI = <*s*>.1 and
A5: QI = GEN(<*s*>, q).1 and
A6: QI1 = GEN(<*s*>, q).(1+1) and
A7: WI-succ_of QI = QI1 by FSM_1:def 2;
  WI = s by A4;
  hence thesis by A2,A3,A5,A6,A7,FINSEQ_1:44;
end;
