reserve n for non zero Nat,
  j,k,l,m for Nat,
  g,h,i for Integer;

theorem Th11:
  for n be non zero Nat, l,m be Nat st l + m <= 2 to_power n - 1
  holds Absval(n-BinarySequence(l) + n-BinarySequence(m)) = l + m
proof
  let n be non zero Nat, l,m be Nat such that
A1: l + m <= 2 to_power n - 1;
A2: l < 2 to_power n by A1,Th8;
  set L = n-BinarySequence(l), M = n-BinarySequence(m);
  add_ovfl(L,M) = FALSE by A1,Th10;
  then L,M are_summable by BINARITH:def 7;
  then
A3: Absval(L+M) = Absval(L) + Absval(M) by BINARITH:22
    .= l + Absval(M) by A2,BINARI_3:35;
  m < 2 to_power n by A1,Th8;
  hence thesis by A3,BINARI_3:35;
end;
