
theorem
  for A being Preorder,
    x,y being Element of A holds
      x =~ y implies (proj A).x = (proj A).y
proof
  let A be Preorder;
  let x,y be Element of A;
  assume A1: x =~ y;
  then A2: [x,y] in EqRelOf A by Def6;
  A3: x in the carrier of A & y in the carrier of A by A1, Th29;
  thus (proj A).x = Class(EqRelOf A, x) by Def8
    .= Class(EqRelOf A, y) by A2, A3, EQREL_1:35
    .= (proj A).y by Def8;
end;
