reserve n for Nat;

theorem Th12:
  for T being non empty 1-sorted, F, G being SetSequence of the
  carrier of T, A being Subset of T st G is subsequence of F & for i being Nat
  holds F.i = A holds G = F
proof
  let T be non empty 1-sorted, F, G be SetSequence of the carrier of T, A be
  Subset of T;
  assume that
A1: G is subsequence of F and
A2: for i being Nat holds F.i = A;
  consider NS being increasing sequence of NAT such that
A3: G = F * NS by A1,VALUED_0:def 17;
  for i being Nat holds G.i = F.i
  proof
    let i be Nat;
     reconsider i as Element of NAT by ORDINAL1:def 12;
    dom NS = NAT by FUNCT_2:def 1;
    then G.i = F.(NS.i) by A3,FUNCT_1:13
      .= A by A2
      .= F.i by A2;
    hence thesis;
  end;
  then
A4: for x being object st x in dom F holds F.x = G.x;
  NAT = dom G & NAT = dom F by FUNCT_2:def 1;
  hence thesis by A4,FUNCT_1:2;
end;
