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