
theorem Th22:
  for L1 be continuous lower-bounded sup-Semilattice for T be
Scott TopAugmentation of L1 for b be Basis of T holds { wayabove inf u where u
  is Subset of T : u in b } is Basis of T
proof
  let L1 be continuous lower-bounded sup-Semilattice;
  let T be Scott TopAugmentation of L1;
  let b be Basis of T;
  set b2 = { wayabove inf u where u is Subset of T : u in b };
  b2 c= bool the carrier of T
  proof
    let z be object;
    assume z in b2;
    then ex u be Subset of T st z = wayabove inf u & u in b;
    hence thesis;
  end;
  then reconsider b2 as Subset-Family of T;
  reconsider b1 = the set of all  wayabove x where x is Element of T as
  Basis of T by WAYBEL11:45;
A1: now
    let A be Subset of T;
    assume
A2: A is open;
    let a be Point of T;
    assume a in A;
    then consider C be Subset of T such that
A3: C in b1 and
A4: a in C and
A5: C c= A by A2,YELLOW_9:31;
    C is open by A3,YELLOW_8:10;
    then consider D be Subset of T such that
A6: D in b and
A7: a in D and
A8: D c= C by A4,YELLOW_9:31;
    D is open by A6,YELLOW_8:10;
    then consider E be Subset of T such that
A9: E in b1 and
A10: a in E and
A11: E c= D by A7,YELLOW_9:31;
    consider z be Element of T such that
A12: E = wayabove z by A9;
    take u = wayabove inf D;
    thus u in b2 by A6;
    reconsider a1 = a as Element of T;
    consider x be Element of T such that
A13: C = wayabove x by A3;
    z << a1 by A10,A12,WAYBEL_3:8;
    then consider y be Element of T such that
A14: z << y and
A15: y << a1 by WAYBEL_4:52;
    inf D is_<=_than D & y in wayabove z by A14,WAYBEL_3:8,YELLOW_0:33;
    then inf D <= y by A11,A12,LATTICE3:def 8;
    then inf D << a1 by A15,WAYBEL_3:2;
    hence a in u by WAYBEL_3:8;
A16: wayabove x c= uparrow x by WAYBEL_3:11;
    ex_inf_of uparrow x,T & ex_inf_of D,T by YELLOW_0:17;
    then inf uparrow x <= inf D by A13,A8,A16,XBOOLE_1:1,YELLOW_0:35;
    then x <= inf D by WAYBEL_0:39;
    then wayabove inf D c= C by A13,WAYBEL_3:12;
    hence u c= A by A5;
  end;
  b2 c= the topology of T
  proof
    let z be object;
    assume z in b2;
    then consider u be Subset of T such that
A17: z = wayabove inf u and
    u in b;
    wayabove inf u is open by WAYBEL11:36;
    hence thesis by A17,PRE_TOPC:def 2;
  end;
  hence thesis by A1,YELLOW_9:32;
end;
