reserve i,j,k,n for Nat;
reserve x,y,z for Tuple of n, BOOLEAN;

theorem Th2:
  for n,k1,k2 being Nat holds n <> 0 & k1 mod n = k2 mod
  n & k1 <= k2 implies ex t being Nat st k2 - k1 = n*t
proof
  let n,k1,k2 be Nat;
  assume that
A1: n <> 0 and
A2: k1 mod n = k2 mod n and
A3: k1 <= k2;
  consider t being Integer such that
A4: t = (k2 div n) - (k1 div n);
  (k2 div n) >= (k1 div n) by A3,NAT_2:24;
  then (k2 div n) - (k1 div n) >= (k1 div n) - (k1 div n) by XREAL_1:9;
  then reconsider t as Element of NAT by A4,INT_1:3;
  take t;
  k1 = n * (k1 div n) + (k1 mod n) & k2 = n * (k2 div n) + (k2 mod n) by A1,
NAT_D:2;
  hence thesis by A2,A4;
end;
