reserve D for non empty set,
  f for FinSequence of D,
  p, p1, p2, p3, q for Element of D,
  i, j, k, l, n for Nat;

theorem
  for f,g,h being FinSequence holds len g = len h & len g < i & i <= len
  (g^f) implies (g^f).i = (h^f).i
proof
  let f,g,h be FinSequence;
  assume that
A1: len g = len h and
A2: len g < i and
A3: i <= len (g^f);
  i <= len g + len f by A3,FINSEQ_1:22;
  then
A4: i - len g <= len g + len f - len g by XREAL_1:9;
  set k = i - len g;
A5: len g - len g < i - len g by A2,XREAL_1:9;
  then reconsider k as Element of NAT by INT_1:3;
  0 + 1 <= i - len g by A5,INT_1:7;
  then
A6: k in dom f by A4,FINSEQ_3:25;
  (g^f).i = (g^f).(k + len g) .= f.k by A6,FINSEQ_1:def 7
    .= (h^f).(len h + k) by A6,FINSEQ_1:def 7;
  hence thesis by A1;
end;
