
theorem
  for a be non weightless Integer, b be non weightless odd Integer st
    a,b are_coprime holds (b + a) mod b <> (b - a) mod b
  proof
    let a be non weightless Integer, b be non weightless odd Integer such that
    A1: a,b are_coprime;
    A1a: not b divides a by A1,N0344; then
    reconsider t = a mod b as non zero Integer by INT162;
    A2: not b divides t by A1a,DIM;
    A3: (b + a) mod b = ((b mod b) + (a mod b)) mod b by NAT_D:66
    .= t mod b;
    A4: (b - a) mod b = ((b mod b) - (a mod b)) mod b by INT_6:7
    .= (-t) mod b;
    t mod b is odd iff (-t) mod b is even by A2,MOO;
    hence thesis by A3,A4;
  end;
