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 :: Theorem 1.14 (1) implies (3 second conjunct) p. 107
  L is continuous implies InclPoset sigma L is continuous
proof
  assume
A1: L is continuous;
  set IPs = InclPoset the topology of L;
A2: the carrier of IPs = the topology of L by YELLOW_1:1;
A3: sigma L = the topology of L by Th23;
  IPs is satisfying_axiom_of_approximation
  proof
    let V be Element of IPs;
    set VV = {wayabove x where x is Element of L : x in V};
    set wV = waybelow V;
    V in sigma L by A3,A2;
    then
A4: V = union VV by A1,Th33;
    now
      let x be object;
      hereby
        assume x in V;
        then consider xU being set such that
A5:     x in xU and
A6:     xU in VV by A4,TARSKI:def 4;
        consider y being Element of L such that
A7:     xU = wayabove y and
A8:     y in V by A6;
        wayabove y is open by A1,WAYBEL11:36;
        then reconsider xU as Element of IPs by A2,A7,PRE_TOPC:def 2;
        xU << V
        proof
          let D be non empty directed Subset of IPs;
          assume V <= sup D;
          then V c= sup D by YELLOW_1:3;
          then V c= union D by YELLOW_1:22;
          then consider DD being set such that
A9:       y in DD and
A10:      DD in D by A8,TARSKI:def 4;
          DD in sigma L by A3,A2,A10;
          then reconsider DD as Subset of L;
          DD is open by A2,A10,PRE_TOPC:def 2;
          then DD is upper by WAYBEL11:def 4;
          then
A11:      uparrow y c= DD by A9,WAYBEL11:42;
          reconsider d = DD as Element of IPs by A10;
          take d;
          thus d in D by A10;
          wayabove y c= uparrow y by WAYBEL_3:11;
          then wayabove y c= DD by A11;
          hence thesis by A7,YELLOW_1:3;
        end;
        then xU in wV;
        hence x in union wV by A5,TARSKI:def 4;
      end;
      assume x in union wV;
      then consider X being set such that
A12:  x in X and
A13:  X in wV by TARSKI:def 4;
      reconsider X as Element of IPs by A13;
      X << V by A13,WAYBEL_3:7;
      then X <= V by WAYBEL_3:1;
      then X c= V by YELLOW_1:3;
      hence x in V by A12;
    end;
    then V = union waybelow V by TARSKI:2;
    hence thesis by YELLOW_1:22;
  end;
  hence thesis by Th23;
end;
