
theorem Th25: :: REMARK 4.20
  for L be non empty Poset for X be Subset of L st X is
  order-generating holds Irr L c= X
proof
  let L be non empty Poset;
  let X be Subset of L;
  assume
A1: X is order-generating;
  let x be object;
  assume
A2: x in Irr L;
  then reconsider x1 = x as Element of L;
A3: x1 = "/\" ((uparrow x1) /\ X,L) & ex_inf_of (uparrow x1) /\ X,L by A1,
WAYBEL_6:def 5;
  x1 is completely-irreducible by A2,Def4;
  then x1 in (uparrow x1) /\ X by A3,Th24;
  hence thesis by XBOOLE_0:def 4;
end;
