reserve X for set, x,y,z for object,
  k,l,n for Nat,
  r for Real;
reserve i,i0,i1,i2,i3,i4,i5,i8,i9,j for Integer;

theorem
  i4 * i5 = i3 implies (i1,i2 are_congruent_mod i3 implies
  i1,i2 are_congruent_mod i4)
proof
  assume
A1: i4 * i5 = i3;
  assume i1,i2 are_congruent_mod i3;
  then consider i0 such that
A2: i3 * i0 = i1 - i2;
  i1 - i2 = i4 * (i5 * i0) by A1,A2;
  hence thesis;
end;
