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
  GEN(<*s1,s2,s3*>, q) = <*q, (the Tran of S).[q, s1],
  (the Tran of S).[(the Tran of S).[q, s1], s2],
  (the Tran of S).[(the Tran of S).[(the Tran of S).[q, s1], s2],s3] *>
proof
  set w = <*s1,s2,s3*>;
  reconsider w1 = <*s1,s2*>, w2 = <*s3*> as FinSequence of I;
  set Q = (the Tran of S).[(the Tran of S).[q,s1], s2];
A1: w = w1^w2 by FINSEQ_1:43;
A2: len w1 = 2 by FINSEQ_1:44;
  GEN(w1, q) = <*q, (the Tran of S).[q, s1], Q*> by Th2;
  then GEN(w1, q).(len w1 + 1) = Q by A2;
  then q,w1-leads_to Q;
  then
A3: GEN(w, q) = Del(GEN(w1,q),len w1 + 1)^GEN(w2,Q) by A1,FSM_1:8;
  Del(GEN(w1,q),len w1 + 1)
  = Del(<*q, (the Tran of S).[q,s1], Q*>, 3) by A2,Th2
    .= <*q, (the Tran of S).[q,s1]*> by WSIERP_1:19;
  then GEN(w, q) = <*q, (the Tran of S).[q,s1]*> ^
  <* Q, (the Tran of S).[Q,s3]*> by A3,Th1;
  hence thesis by FINSEQ_4:74;
end;
