
theorem Th16:
  for N being meet-continuous Lawson complete TopLattice for S
being Scott TopAugmentation of N for x being Point of S, y being Point of N for
J being Basis of y st x = y holds {uparrow A where A is Subset of N: A in J} is
  Basis of x
proof
  let N be meet-continuous Lawson complete TopLattice, S be Scott
  TopAugmentation of N, x be Point of S, y be Point of N, J be Basis of y such
  that
A1: x = y;
  set Z = {uparrow A where A is Subset of N: A in J};
  set K = Z;
A2: the RelStr of N = the RelStr of S by YELLOW_9:def 4;
  K c= bool the carrier of S
  proof
    let k be object;
    assume k in K;
    then ex A being Subset of N st k = uparrow A & A in J;
    hence thesis by A2;
  end;
  then reconsider K as Subset-Family of S;
  K is Basis of x
  proof
A3:  K is open
    proof
      let k be Subset of S;
      assume k in K;
      then consider A being Subset of N such that
A4:   k = uparrow A and
A5:   A in J;
      reconsider A9 = A as Subset of S by A2;
      uparrow A9 is open by A5,Th15,YELLOW_8:12;
      then uparrow A9 in the topology of S;
      then uparrow A in the topology of S by A2,WAYBEL_0:13;
      hence thesis by A4;
    end;
    K is x-quasi_basis
    proof
    for k being set st k in K holds x in k
    proof
      let k be set;
      assume k in K;
      then consider A being Subset of N such that
A6:   k = uparrow A and
A7:   A in J;
A8:   A c= uparrow A by WAYBEL_0:16;
      y in Intersect J by YELLOW_8:def 1;
      then y in A by A7,SETFAM_1:43;
      hence thesis by A8,A1,A6;
    end;
    hence x in Intersect K by SETFAM_1:43;
    let sA be Subset of S such that
A9: sA is open and
A10: x in sA;
    sA is upper by A9,WAYBEL11:def 4;
    then
A11: uparrow sA c= sA by WAYBEL_0:24;
    reconsider lA = sA as Subset of N by A2;
    N is Lawson correct TopAugmentation of S by A2,YELLOW_9:def 4;
    then lA is open by A9,WAYBEL19:37;
    then lA in lambda N by Th12;
    then
A12: uparrow lA in sigma S by Th14;
A13: lA c= uparrow lA by WAYBEL_0:16;
    sigma N c= lambda N by Th10;
    then sigma S c= lambda N by A2,YELLOW_9:52;
    then uparrow lA is open by A12,Th12;
    then consider lV1 being Subset of N such that
A14: lV1 in J and
A15: lV1 c= uparrow lA by A13,A1,A10,YELLOW_8:def 1;
    reconsider sUV = uparrow lV1 as Subset of S by A2;
    take sUV;
    thus sUV in K by A14;
A16: lV1 is_coarser_than uparrow lA by A15,WAYBEL12:16;
    sA c= uparrow sA by WAYBEL_0:16;
    then
A17: sA = uparrow sA by A11;
    uparrow sA = uparrow lA by A2,WAYBEL_0:13;
    hence thesis by A16,A17,YELLOW12:28;
  end;
  hence thesis by A3;
  end;
  hence thesis;
end;
