
theorem Th13:
  for N being meet-continuous LATTICE, A being Subset of N st A is
  property(S) holds uparrow A is property(S)
proof
  let N be meet-continuous LATTICE, A be Subset of N such that
A1: for D being non empty directed Subset of N st sup D in A ex y being
Element of N st y in D & for x being Element of N st x in D & x >= y holds x in
  A;
  let D be non empty directed Subset of N;
  assume sup D in uparrow A;
  then consider a being Element of N such that
A2: a <= sup D and
A3: a in A by WAYBEL_0:def 16;
  reconsider aa = {a} as non empty directed Subset of N by WAYBEL_0:5;
  a = sup ({a} "/\" D) by A2,WAYBEL_2:52;
  then consider y being Element of N such that
A4: y in aa "/\" D and
A5: for x being Element of N st x in aa "/\" D & x >= y holds x in A by A1,A3;
  aa "/\" D = {a "/\" d where d is Element of N: d in D} by YELLOW_4:42;
  then consider d being Element of N such that
A6: y = a "/\" d and
A7: d in D by A4;
  take d;
  thus d in D by A7;
  let x be Element of N such that
  x in D and
A8: x >= d;
  d >= y by A6,YELLOW_0:23;
  then
A9: x >= y by A8,ORDERS_2:3;
  y in A by A4,A5,ORDERS_2:1;
  hence thesis by A9,WAYBEL_0:def 16;
end;
