reserve L for complete Scott TopLattice,
  x for Element of L,
  X, Y for Subset of L,
  V, W for Element of InclPoset sigma L,
  VV for Subset of InclPoset sigma L;

theorem :: Corollary 1.15 (3) implies (2) p. 108
:: The proof of ((3) implies (1)) is split into two parts
:: This one proves ((3) implies (2)) and the next is ((2) implies (1)).
  InclPoset sigma L is algebraic & (for V ex VV st V = sup VV & for W st
W in VV holds W is co-prime) implies ex B being Basis of L st B = {uparrow x :
  x in the carrier of CompactSublatt L}
proof
  set IPt = InclPoset the topology of L;
  set IPs = InclPoset sigma L;
A1: the carrier of IPs = sigma L by YELLOW_1:1;
  set B = {uparrow k where k is Element of L : k in the carrier of
  CompactSublatt L};
  B c= bool the carrier of L
  proof
    let x be object;
    assume x in B;
    then ex k being Element of L st x = uparrow k & k in the carrier of
    CompactSublatt L;
    hence thesis;
  end;
  then reconsider B as Subset-Family of L;
  assume that
A2: InclPoset sigma L is algebraic and
A3: for V being Element of InclPoset sigma L ex X being Subset of
InclPoset sigma L st V = sup X & for x being Element of InclPoset sigma L st x
  in X holds x is co-prime;
  IPs = IPt by Th23;
  then reconsider ips = InclPoset sigma L as algebraic LATTICE by A2;
  reconsider B as Subset-Family of L;
A4: B c= the topology of L
  proof
    let x be object;
    assume x in B;
    then consider k being Element of L such that
A5: x = uparrow k and
A6: k in the carrier of CompactSublatt L;
    k is compact by A6,WAYBEL_8:def 1;
    then uparrow k is Open by WAYBEL_8:2;
    then uparrow k is open by WAYBEL11:41;
    hence thesis by A5,PRE_TOPC:def 2;
  end;
A7: sigma L = the topology of L by Th23;
  ips is continuous & for x being Element of L ex X being Basis of x st
  for Y being Subset of L st Y in X holds Y is open filtered by A3,Th39;
  then
  for x being Element of L holds x = "\/" ({inf V where V is Subset of L :
  x in V & V in sigma L}, L) by Th37;
  then
A8: L is continuous by Th38;
  now
    let x be Point of L;
    set Bx = {uparrow k where k is Element of L : uparrow k in B & x in
    uparrow k};
    Bx c= bool the carrier of L
    proof
      let y be object;
      assume y in Bx;
      then ex k being Element of L st y = uparrow k & uparrow k in B & x in
      uparrow k;
      hence thesis;
    end;
    then reconsider Bx as Subset-Family of L;
    reconsider Bx as Subset-Family of L;
    Bx is Basis of x
    proof
A9:  Bx is open
      proof
        let y be Subset of L;
        assume y in Bx;
        then ex k being Element of L st y = uparrow k & uparrow k in B & x in
        uparrow k;
        hence thesis by A4,PRE_TOPC:def 2;
      end;
      Bx is x-quasi_basis
      proof
      now
        per cases;
        suppose
          Bx is empty;
          then Intersect Bx = the carrier of L by SETFAM_1:def 9;
          hence x in Intersect Bx;
        end;
        suppose
A10:       Bx is non empty;
A11:      now
            let Y be set;
            assume Y in Bx;
            then
            ex k being Element of L st Y = uparrow k & uparrow k in B & x
            in uparrow k;
            hence x in Y;
          end;
          Intersect Bx = meet Bx by A10,SETFAM_1:def 9;
          hence x in Intersect Bx by A10,A11,SETFAM_1:def 1;
        end;
      end;
      hence x in Intersect Bx;
      let S be Subset of L such that
A12:  S is open and
A13:  x in S;
      reconsider S9 = S as Element of IPt by A7,A1,A12,PRE_TOPC:def 2;
      S9 = sup compactbelow S9 by A2,A7,WAYBEL_8:def 3;
      then S9 = union compactbelow S9 by YELLOW_1:22;
      then consider UA being set such that
A14:  x in UA and
A15:  UA in compactbelow S9 by A13,TARSKI:def 4;
      reconsider UA as Element of IPt by A15;
      UA is compact by A15,WAYBEL_8:4;
      then
A16:  UA << UA;
      UA in the topology of L by A7,A1;
      then reconsider UA9 = UA as Subset of L;
      UA <= S9 by A15,WAYBEL_8:4;
      then
A17:  UA c= S by YELLOW_1:3;
      consider F being Subset of InclPoset sigma L such that
A18:  UA = sup F and
A19:  for x being Element of InclPoset sigma L st x in F holds x is
      co-prime by A3,A7;
      reconsider F9 = F as Subset-Family of L by A1,XBOOLE_1:1;
A20:  UA = union F by A7,A18,YELLOW_1:22;
      F9 is open
      by A7,A1,PRE_TOPC:def 2;
      then consider G being finite Subset of F9 such that
A21:  UA c= union G by A20,A16,WAYBEL_3:34;
      union G c= union F by ZFMISC_1:77;
      then
A22:  UA = union G by A20,A21;
      reconsider G as finite Subset-Family of L by XBOOLE_1:1;
      consider Gg being finite Subset-Family of L such that
A23:  Gg c= G and
A24:  union Gg = union G and
A25:  for g being Subset of L st g in Gg holds not g c= union (Gg\{g} ) by Th1;
      consider U1 being set such that
A26:  x in U1 and
A27:  U1 in Gg by A14,A21,A24,TARSKI:def 4;
A28:  Gg c= F by A23,XBOOLE_1:1;
      then U1 in F by A27;
      then reconsider U1 as Element of IPs;
      U1 in the topology of L by A7,A1;
      then reconsider U19 = U1 as Subset of L;
      set k = inf U19;
A29:  U19 c= uparrow k
      proof
        let x be object;
        assume
A30:    x in U19;
        then reconsider x9 = x as Element of L;
        k is_<=_than U19 by YELLOW_0:33;
        then k <= x9 by A30,LATTICE3:def 8;
        hence thesis by WAYBEL_0:18;
      end;
      U1 is co-prime by A19,A27,A28;
      then
A31:  U19 is filtered upper by Th27;
      now
        set D = {(downarrow u)` where u is Element of L : u in U19};
A32:    D c= the topology of L
        proof
          let d be object;
          assume d in D;
          then consider u being Element of L such that
A33:      d = (downarrow u)` and
          u in U19;
          (downarrow u)` is open by WAYBEL11:12;
          hence thesis by A33,PRE_TOPC:def 2;
        end;
        consider u being set such that
A34:    u in U19 by A26;
        reconsider u as Element of L by A34;
        (downarrow u)` in D by A34;
        then reconsider D as non empty Subset of IPt by A32,YELLOW_1:1;
        assume
A35:    not k in U19;
        now
          assume not UA c= union D;
          then consider l being object such that
A36:      l in UA9 and
A37:      not l in union D;
          reconsider l as Element of L by A36;
          consider Uk being set such that
A38:      l in Uk and
A39:      Uk in Gg by A21,A24,A36,TARSKI:def 4;
A40:      Gg c= F by A23,XBOOLE_1:1;
          then Uk in F by A39;
          then reconsider Uk as Element of IPs;
          Uk in the topology of L by A7,A1;
          then reconsider Uk9 = Uk as Subset of L;
          Uk is co-prime by A19,A39,A40;
          then
A41:      Uk9 is filtered upper by Th27;
          now
            assume not l is_<=_than U19;
            then consider m being Element of L such that
A42:        m in U19 and
A43:        not l <= m by LATTICE3:def 8;
            (downarrow m)` in D by A42;
            then not l in (downarrow m)` by A37,TARSKI:def 4;
            then l in downarrow m by XBOOLE_0:def 5;
            hence contradiction by A43,WAYBEL_0:17;
          end;
          then l <= k by YELLOW_0:33;
          then
A44:      k in Uk9 by A38,A41;
A45:      k is_<=_than U19 by YELLOW_0:33;
A46:      U19 c= Uk
          proof
            let u be object;
            assume
A47:        u in U19;
            then reconsider d = u as Element of L;
            k <= d by A45,A47,LATTICE3:def 8;
            hence thesis by A41,A44;
          end;
          U19 c= union (Gg\{U19})
          proof
            let u be object;
            assume
A48:        u in U19;
            Uk in Gg\{U19} by A35,A39,A44,ZFMISC_1:56;
            hence thesis by A46,A48,TARSKI:def 4;
          end;
          hence contradiction by A25,A27;
        end;
        then UA c= sup D by YELLOW_1:22;
        then
A49:    UA <= sup D by YELLOW_1:3;
        D is directed by A7,A31,Th25;
        then consider d being Element of IPt such that
A50:    d in D and
A51:    UA <= d by A16,A49;
        consider u being Element of L such that
A52:    d = (downarrow u)` and
A53:    u in U19 by A50;
        U1 c= UA by A20,A27,A28,ZFMISC_1:74;
        then
A54:    u in UA by A53;
A55:    u <= u;
        UA c= d by A51,YELLOW_1:3;
        then not u in downarrow u by A52,A54,XBOOLE_0:def 5;
        hence contradiction by A55,WAYBEL_0:17;
      end;
      then uparrow k c= U19 by A31,WAYBEL11:42;
      then
A56:  U19 = uparrow k by A29;
      take V = uparrow k;
      U19 is open by A7,A1,PRE_TOPC:def 2;
      then U19 is Open by A8,A31,WAYBEL11:46;
      then k is compact by A56,WAYBEL_8:2;
      then k in the carrier of CompactSublatt L by WAYBEL_8:def 1;
      then uparrow k in B;
      hence V in Bx by A26,A29;
      U1 c= UA by A22,A24,A27,ZFMISC_1:74;
      hence thesis by A56,A17;
    end;
    hence thesis by A9;
    end;
    then reconsider Bx as Basis of x;
    take Bx;
    thus Bx c= B
    proof
      let y be object;
      assume y in Bx;
      then ex k being Element of L st y = uparrow k & uparrow k in B & x in
      uparrow k;
      hence thesis;
    end;
  end;
  then reconsider B as Basis of L by A4,YELLOW_8:14;
  take B;
  thus thesis;
end;
