reserve x, y, i for object,
  L for up-complete Semilattice;
reserve L for complete LATTICE,
  a, b, c for Element of L,
  J for non empty set,
  K for non-empty ManySortedSet of J;

theorem Th17:
  (L is continuous & for c st c << a holds c <= b) implies a <= b
proof
  assume that
A1: L is continuous and
A2: for c st c << a holds c <= b;
  for c st c in waybelow a holds c <= b by A2,WAYBEL_3:7;
  then waybelow a is_<=_than b;
  then sup waybelow a <= b by YELLOW_0:32;
  hence thesis by A1,WAYBEL_3:def 5;
end;
