reserve X for set;

theorem
  for X be non empty set holds InclPoset X is with_suprema implies for x
  ,y be Element of InclPoset X holds x \/ y c= x "\/" y
proof
  let X be non empty set;
  assume
A1: InclPoset X is with_suprema;
  let x,y be Element of InclPoset X;
  y <= x "\/" y by A1,YELLOW_0:22;
  then
A2: y c= x "\/" y by Th3;
  x <= x "\/" y by A1,YELLOW_0:22;
  then x c= x "\/" y by Th3;
  hence thesis by A2,XBOOLE_1:8;
end;
