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 Th27: :: Proposition 1.11 (i) p. 105
  V = X implies (V is co-prime iff X is filtered upper)
proof
  assume
A1: V = X;
A2: the TopStruct of L = ConvergenceSpace Scott-Convergence L by WAYBEL11:32;
A3: sigma L = the topology of ConvergenceSpace Scott-Convergence L by
WAYBEL11:def 12;
A4: the carrier of InclPoset sigma L = sigma L by YELLOW_1:1;
  then
A5: X is upper by A1,A3,WAYBEL11:31;
  hereby
    assume
A6: V is co-prime;
    thus X is filtered
    proof
      let v, w be Element of L such that
A7:   v in X and
A8:   w in X;
      (downarrow w)` is open & (downarrow v)` is open by WAYBEL11:12;
      then reconsider mdw = (downarrow w)`, mdv = (downarrow v)` as Element of
      InclPoset sigma L by A3,A4,A2,PRE_TOPC:def 2;
      w <= w;
      then w in downarrow w by WAYBEL_0:17;
      then not V c= (downarrow w)` by A1,A8,XBOOLE_0:def 5;
      then
A9:   not V <= mdw by YELLOW_1:3;
      v <= v;
      then v in downarrow v by WAYBEL_0:17;
      then not V c= (downarrow v)` by A1,A7,XBOOLE_0:def 5;
      then not V <= mdv by YELLOW_1:3;
      then not V <= mdv "\/" mdw by A3,A6,A9,Th14;
      then
A10:  not V c= mdv "\/" mdw by YELLOW_1:3;
      take z = v"/\"w;
A11:  mdv \/ mdw = ((downarrow v) /\ downarrow w)` by XBOOLE_1:54
        .= (downarrow(v"/\"w))` by Th3;
      mdv \/ mdw c= mdv "\/" mdw by A3,YELLOW_1:6;
      then not V c= mdv \/ mdw by A10;
      then X meets (downarrow(v"/\"w))`` by A1,A11,SUBSET_1:24;
      then X/\(downarrow(v"/\"w))`` <> {};
      then consider zz being object such that
A12:  zz in X/\downarrow(v"/\"w) by XBOOLE_0:def 1;
A13:  zz in downarrow(v"/\"w) by A12,XBOOLE_0:def 4;
A14:  zz in X by A12,XBOOLE_0:def 4;
      reconsider zz as Element of L by A12;
      zz <= v"/\"w by A13,WAYBEL_0:17;
      hence z in X by A5,A14;
      thus z <= v & z <= w by YELLOW_0:23;
    end;
    thus X is upper by A1,A3,A4,WAYBEL11:31;
  end;
  assume
A15: X is filtered upper;
  assume not V is co-prime;
  then consider Xx, Y being Element of InclPoset sigma L such that
A16: V <= Xx "\/" Y and
A17: not V <= Xx and
A18: not V <= Y by A3,Th14;
  Xx in sigma L & Y in sigma L by A4;
  then reconsider Xx9 = Xx, Y9 = Y as Subset of L;
A19: Y9 is open by A3,A4,A2,PRE_TOPC:def 2;
  then
A20: Y9 is upper by WAYBEL11:def 4;
A21: Xx9 is open by A3,A4,A2,PRE_TOPC:def 2;
  then Xx9 \/ Y9 is open by A19;
  then Xx \/ Y in sigma L by A3,A2,PRE_TOPC:def 2;
  then Xx \/ Y = Xx "\/" Y by YELLOW_1:8;
  then
A22: V c= Xx \/ Y by A16,YELLOW_1:3;
  not V c= Y by A18,YELLOW_1:3;
  then consider w being object such that
A23: w in V and
A24: not w in Y;
  not V c= Xx by A17,YELLOW_1:3;
  then consider v being object such that
A25: v in V and
A26: not v in Xx;
  reconsider v, w as Element of L by A1,A25,A23;
A27: Xx9 is upper by A21,WAYBEL11:def 4;
A28: now
    assume
A29: v"/\"w in Xx9 \/ Y9;
    per cases by A29,XBOOLE_0:def 3;
    suppose
A30:  v"/\"w in Xx9;
      v"/\"w <= v by YELLOW_0:23;
      hence contradiction by A26,A27,A30;
    end;
    suppose
A31:  v"/\"w in Y9;
      v"/\"w <= w by YELLOW_0:23;
      hence contradiction by A24,A20,A31;
    end;
  end;
  v"/\"w in X by A1,A15,A25,A23,WAYBEL_0:41;
  hence contradiction by A1,A22,A28;
end;
