reserve i, j, n for Element of NAT,
  f, g, h, k for FinSequence of REAL,
  M, N for non empty MetrSpace;

theorem Th4:
  for i being Nat for f, g being FinSequence st len f < i & i <=
  len f + len g holds i - len f in dom g
proof
  let i be Nat;
  let f, g be FinSequence such that
A1: len f < i and
A2: i <= len f + len g;
A3: i-len f is Element of NAT by A1,INT_1:5;
A4: i-len f <= len f + len g - len f by A2,XREAL_1:9;
  i-len f > len f-len f by A1,XREAL_1:14;
  then 1 <= i-len f by A3,NAT_1:14;
  hence thesis by A3,A4,FINSEQ_3:25;
end;
