reserve D for non empty set,
  i,j,k for Nat,
  n,m for Nat,
  r for Real,
  e for real-valued FinSequence;

theorem Th35:
  for e1,e2 being FinSequence of REAL for f1,f2 being FinSequence
of F_Real st len e1 = len e2 & e1 = f1 & e2 = f2 holds mlt(e1,e2) = mlt(f1,f2)
proof
  let e1,e2 being FinSequence of REAL;
  let f1,f2 being FinSequence of F_Real such that
A1: len e1 = len e2 and
A2: e1 = f1 and
A3: e2 = f2;
  set l = len e1;
  set Z = { f where f is Element of (the carrier of F_Real)*: len f = l};
  f1 is Element of (the carrier of F_Real)* by FINSEQ_1:def 11;
  then f1 in Z by A2;
  then reconsider f3=f1 as Element of l-tuples_on the carrier of F_Real;
  f2 is Element of (the carrier of F_Real)* by FINSEQ_1:def 11;
  then f2 in Z by A1,A3;
  then reconsider f4=f2 as Element of l-tuples_on the carrier of F_Real;
  set Y = { e where e is Element of REAL*: len e = l};
  e2 is Element of REAL* by FINSEQ_1:def 11;
  then e2 in Y by A1;
  then reconsider e4=e2 as Element of l-tuples_on REAL;
  reconsider e3=e1 as Element of l-tuples_on REAL by FINSEQ_2:92;
  mlt(e3,e4) = mlt(f3,f4) by A2,A3,Th34;
  hence thesis;
end;
