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

theorem Th10:
  l + m <= 2 to_power n - 1 implies add_ovfl(n-BinarySequence(l),n
  -BinarySequence(m)) = FALSE
proof
  set L = n-BinarySequence(l), M = n-BinarySequence(m);
A1: Absval(L+M) + 2 to_power n >= 2 to_power n by NAT_1:11;
  assume
A2: l + m <= 2 to_power n - 1;
  then
A3: l < 2 to_power n by Th8;
  assume add_ovfl(L,M) <> FALSE;
  then
A4: IFEQ(add_ovfl(L,M),FALSE,0,2 to_power n) = 2 to_power n by FUNCOP_1:def 8;
A5: m < 2 to_power n by A2,Th8;
  Absval(L+M) + IFEQ(add_ovfl(L,M),FALSE,0,2 to_power n) = Absval(L)+
  Absval(M) by BINARITH:21
    .= l + Absval(M) by A3,BINARI_3:35
    .= l + m by A5,BINARI_3:35;
  hence contradiction by A2,A4,A1,XREAL_1:146,XXREAL_0:2;
end;
