
theorem Th45:
  for A being non empty Preorder, x, y being Element of A st x <= y holds
   (proj A).x <= (proj A).y
proof
  let A be non empty Preorder;
  let x,y be Element of A;
  assume A1: x <= y;
  reconsider X = Class(EqRelOf A, x) as Element of Class EqRelOf A
    by EQREL_1:def 3;
  reconsider Y = Class(EqRelOf A, y) as Element of Class EqRelOf A
    by EQREL_1:def 3;
  A2: [X,Y] in the InternalRel of QuotientOrder(A) by A1, Def7;
  X = (proj A).x & Y = (proj A).y by Def8;
  hence thesis by A2, ORDERS_2:def 5;
end;
