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 Th26:
  X is open & x in X implies inf X << x
proof
  assume that
A1: X is open and
A2: x in X;
A3: X is upper property(S) by A1,WAYBEL11:15;
  now
    let D be non empty directed Subset of L;
    assume x <= sup D;
    then sup D in X by A2,A3;
    then consider y being Element of L such that
A4: y in D and
A5: for x being Element of L st x in D & x >= y holds x in X by A3,
WAYBEL11:def 3;
    take y;
    thus y in D by A4;
    y <= y;
    then inf X is_<=_than X & y in X by A4,A5,YELLOW_0:33;
    hence inf X <= y by LATTICE3:def 8;
  end;
  hence thesis;
end;
