
theorem
  for L being up-complete Semilattice
  st for x being Element of L holds waybelow x is non empty directed holds
  L is satisfying_axiom_of_approximation iff
  for x,y being Element of L st not x <= y
  ex u being Element of L st u << x & not u <= y
proof
  let L be up-complete Semilattice such that
A1: for x being Element of L holds waybelow x is non empty directed;
  hereby
    assume
A2: L is satisfying_axiom_of_approximation;
    let x,y be Element of L;
    assume
A3: not x <= y;
A4: waybelow x is non empty directed by A1;
A5: x = sup waybelow x by A2;
    ex_sup_of waybelow x,L by A4,WAYBEL_0:75;
    then y is_>=_than waybelow x implies y >= x by A5,YELLOW_0:def 9;
    then consider u being Element of L such that
A6: u in waybelow x and
A7: not u <= y by A3;
    take u;
    thus u << x & not u <= y by A6,A7,Th7;
  end;
  assume
A8: for x,y being Element of L st not x <= y
  ex u being Element of L st u << x & not u <= y;
  let x be Element of L;
  waybelow x is non empty directed by A1;
  then
A9: ex_sup_of waybelow x,L by WAYBEL_0:75;
A10: x is_>=_than waybelow x by Th9;
  now
    let y be Element of L;
    assume that
A11: y is_>=_than waybelow x and
A12: not x <= y;
    consider u being Element of L such that
A13: u << x and
A14: not u <= y by A8,A12;
    u in waybelow x by A13;
    hence contradiction by A11,A14;
  end;
  hence thesis by A9,A10,YELLOW_0:def 9;
end;
