
theorem Th35:
  for N being meet-continuous Lawson complete TopLattice holds N
is continuous iff N is with_open_semilattices & InclPoset sigma N is continuous
proof
  let N be meet-continuous Lawson complete TopLattice;
  set S = the Scott TopAugmentation of N;
A1: the RelStr of N = the RelStr of S by YELLOW_9:def 4;
  hereby
    assume
A2: N is continuous;
    for x being Point of S ex J being Basis of x st for W being Subset of
    S st W in J holds W is Filter of S
    proof
      let x be Point of S;
      consider J being Basis of x such that
A3:   for W being Subset of S st W in J holds W is open filtered by A2,
WAYBEL14:35;
      take J;
      let W be Subset of S;
      assume
A4:   W in J;
      then W is open by A3;
      hence thesis by A3,A4,WAYBEL11:def 4,YELLOW_8:12;
    end;
    hence N is with_open_semilattices by Th32;
    InclPoset sigma S is continuous by A2,WAYBEL14:36;
    hence InclPoset sigma N is continuous by A1,YELLOW_9:52;
  end;
  assume that
A5: N is with_open_semilattices and
A6: InclPoset sigma N is continuous;
A7: for x being Element of S ex J being Basis of x st for Y being Subset of
  S st Y in J holds Y is open filtered
  proof
    let x be Element of S;
    reconsider y = x as Element of N by A1;
    consider J being Basis of y such that
A8: for A being Subset of N st A in J holds subrelstr A is
    meet-inheriting by A5;
    reconsider J9 = {uparrow A where A is Subset of N: A in J} as Basis of x
    by Th16;
    take J9;
    let Y be Subset of S;
    assume
A9: Y in J9;
    then consider A being Subset of N such that
A10: Y = uparrow A and
A11: A in J;
    subrelstr A is meet-inheriting by A8,A11;
    then A is filtered by YELLOW12:26;
    hence thesis by A1,A9,A10,WAYBEL_0:4,YELLOW_8:12;
  end;
  sigma S = sigma N by A1,YELLOW_9:52;
  then
  for x being Element of S holds x = "\/" ({inf X where X is Subset of S:
  x in X & X in sigma S}, S) by A7,A6,WAYBEL14:37;
  then S is continuous by WAYBEL14:38;
  hence thesis by A1,YELLOW12:15;
end;
