
theorem MAB:
  for a,b be Integer, c be non zero Nat holds
  ((a+b) mod c = b mod c implies a mod c = 0) &
  ((a+b) mod c <> b mod c implies a mod c > 0)
  proof
    let a,b be Integer, c be non zero Nat;
    L1: (a+b) mod c = b mod c implies a mod c = 0
    proof
      assume (a+b) mod c = b mod c; then
      0 = ((a+b) mod c - (b mod c)) mod c
      .= (a+b - b) mod c by INT_6:7;
      hence thesis;
    end;
    (a+b) mod c = ((a mod c) + (b mod c)) mod c by NAT_D:66; then
    a mod c = 0 implies (a+b) mod c = b mod c;
    hence thesis by L1,INT_1:57;
  end;
