reserve a,b,m,x,y,i1,i2,i3,i for Integer,
  k,p,q,n for Nat,
  c,c1,c2 for Element of NAT,
  z for set;

theorem
  for a,b,m being Nat st m > 0 holds a mod m = b mod m iff m divides (a- b)
proof
  let a,b,m be Nat;
  assume
A1: m > 0;
  thus a mod m = b mod m implies m divides (a-b)
  proof
    assume a mod m = b mod m;
    then (a - b) mod m = 0 by A1,Th22;
    hence thesis by A1,INT_1:62;
  end;
  assume m divides (a - b);
  then (a - b) mod m = 0 by A1,INT_1:62;
  hence thesis by A1,Th22;
end;
