reserve m, n, i, k for Nat;
reserve IAlph, OAlph for non empty set,
  fsm for non empty FSM over IAlph,
  s for Element of IAlph,
  w, w1, w2 for FinSequence of IAlph,
  q, q9, q1, q2 for State of fsm;
reserve tfsm, tfsm1, tfsm2, tfsm3 for non empty Mealy-FSM over IAlph, OAlph,
  sfsm for non empty Moore-FSM over IAlph, OAlph,
  qs for State of sfsm,
  q, q1, q2 , q3, qa, qb, qc, qa9, qt, q1t, q2t for State of tfsm,
  q11, q12 for State of tfsm1,
  q21, q22 for State of tfsm2;

theorem Th12:
  q11, w1 -leads_to q12 & q21, w1 -leads_to q22 & (q12,w2)
  -response <> (q22,w2)-response implies (q11,w1^w2)-response <> (q21,w1^w2)
  -response
proof
  assume that
A1: q11, w1 -leads_to q12 and
A2: q21, w1 -leads_to q22 and
A3: (q12,w2)-response <> (q22,w2)-response;
  set r12 = (q12,w2)-response, r22 = (q22,w2)-response;
A4: len r22 = len w2 by Def6;
  set w = w1 ^ w2;
  set r1w1 = (q11,w1)-response, r2w1 = (q21,w1)-response;
  assume
A5: (q11,w1^w2)-response = (q21,w1^w2)-response;
  set r21 = (q21,w)-response;
A6: r21 = r2w1 ^ r22 by A2,Th11;
  set r11 = (q11,w)-response;
A7: r11 = r1w1 ^ r12 by A1,Th11;
A8: len r1w1 = len w1 by Def6;
A9: len r12 = len w2 by Def6;
  then
A10: dom w2 = Seg len r12 by FINSEQ_1:def 3;
  then dom w2 = dom r12 by FINSEQ_1:def 3;
  then consider j being Nat such that
A11: j in dom w2 and
A12: r12.j <> r22.j by A3,A9,A4,FINSEQ_2:9;
A13: len r2w1 = len w1 by Def6;
  j in dom r12 by A10,A11,FINSEQ_1:def 3;
  then
A14: r11.(len w1 + j)=r12.j by A8,A7,FINSEQ_1:def 7;
  j in dom r22 by A9,A4,A10,A11,FINSEQ_1:def 3;
  hence contradiction by A5,A13,A12,A6,A14,FINSEQ_1:def 7;
end;
