
theorem Th10:
  for L1 be continuous lower-bounded sup-Semilattice for T1 be
Scott TopAugmentation of L1 for T2 be Lawson correct TopAugmentation of L1 for
  B2 be Basis of T2 holds { uparrow V where V is Subset of T2 : V in B2 } is
  Basis of T1
proof
  let L1 be continuous lower-bounded sup-Semilattice;
  let T1 be Scott TopAugmentation of L1;
  let T2 be Lawson correct TopAugmentation of L1;
  let B2 be Basis of T2;
A1: the RelStr of T1 = the RelStr of L1 & the RelStr of T2 = the RelStr of
  L1 by YELLOW_9:def 4;
  { uparrow V where V is Subset of T2 : V in B2 } c= bool the carrier of T1
  proof
    let z be object;
    assume z in { uparrow V where V is Subset of T2 : V in B2 };
    then ex V be Subset of T2 st z = uparrow V & V in B2;
    hence thesis by A1;
  end;
  then reconsider upV = { uparrow V where V is Subset of T2 : V in B2 } as
  Subset-Family of T1;
  reconsider upV as Subset-Family of T1;
A2: the topology of T1 c= UniCl upV
  proof
    let z be object;
    assume
A3: z in the topology of T1;
    then reconsider z2 = z as Subset of T1;
    z2 is open by A3,PRE_TOPC:def 2;
    then z2 is upper by WAYBEL11:def 4;
    then
A4: uparrow z2 c= z2 by WAYBEL_0:24;
    reconsider z1 = z as Subset of T2 by A1,A3;
    z in sigma T1 by A3,WAYBEL14:23;
    then sigma T2 c= lambda T2 & z in sigma T2 by A1,WAYBEL30:10,YELLOW_9:52;
    then z in lambda T2;
    then z in the topology of T2 by WAYBEL30:9;
    then
A5: z1 is open by PRE_TOPC:def 2;
    { uparrow G where G is Subset of T2 : G in B2 & G c= z1 } c= bool the
    carrier of T1
    proof
      let v be object;
      assume v in { uparrow G where G is Subset of T2 : G in B2 & G c= z1 };
      then ex G be Subset of T2 st v = uparrow G & G in B2 & G c= z1;
      hence thesis by A1;
    end;
    then reconsider
    Y = { uparrow G where G is Subset of T2 : G in B2 & G c= z1 }
    as Subset-Family of T1;
    { G where G is Subset of T2 : G in B2 & G c= z1 } c= bool the carrier of T1
    proof
      let v be object;
      assume v in { G where G is Subset of T2 : G in B2 & G c= z1 };
      then ex G be Subset of T2 st v = G & G in B2 & G c= z1;
      hence thesis by A1;
    end;
    then reconsider Y1 = { G where G is Subset of T2 : G in B2 & G c= z1 } as
    Subset-Family of T1;
    defpred P[set] means $1 in B2 & $1 c= z1;
    reconsider Y as Subset-Family of T1;
    reconsider Y1 as Subset-Family of T1;
    reconsider Y3 = Y1 as Subset-Family of T2 by A1;
A6: Y c= upV
    proof
      let v be object;
      assume v in Y;
      then ex G be Subset of T2 st v = uparrow G & G in B2 & G c= z1;
      hence thesis;
    end;
A7: for S be Subset-Family of T2 st S = { X where X is Subset of T2 : P[X
    ]} holds uparrow union S = union { uparrow X where X is Subset of T2: P[X]}
    from UparrowUnion;
    z2 c= uparrow z2 by WAYBEL_0:16;
    then z1 = uparrow z2 by A4,XBOOLE_0:def 10
      .= uparrow union Y1 by A5,YELLOW_8:9
      .= uparrow union Y3 by A1,WAYBEL_0:13
      .= union Y by A7;
    hence thesis by A6,CANTOR_1:def 1;
  end;
  upV c= the topology of T1
  proof
    let z be object;
    assume z in upV;
    then consider V be Subset of T2 such that
A8: z = uparrow V and
A9: V in B2;
A10: T1 is Scott TopAugmentation of T2 by A1,YELLOW_9:def 4;
    B2 c= the topology of T2 by TOPS_2:64;
    then V in the topology of T2 by A9;
    then V in lambda T2 by WAYBEL30:9;
    then uparrow V in sigma T1 by A10,WAYBEL30:14;
    hence thesis by A8,WAYBEL14:23;
  end;
  hence thesis by A2,CANTOR_1:def 2,TOPS_2:64;
end;
