
theorem
  for a,b be Nat,c be non zero Nat holds
  (a mod c)*(b mod c) >= c implies a mod c > 1
  proof
    let a,b be Nat,c be non zero Nat;
    assume
    A1: (a mod c)*(b mod c) >= c;
    a mod c > 1 or (a mod c)*(b mod c) <= 1*(b mod c) by XREAL_1:64; then
    (a mod c) > 1 or (b mod c) >= c by A1,XXREAL_0:2;
    hence thesis by INT_1:58;
  end;
