reserve x, y, i for object,
  L for up-complete Semilattice;

theorem ::Theorem 2.1 (4) implies (1)
  for L being up-complete lower-bounded LATTICE holds SupMap L is
  upper_adjoint implies L is continuous
proof
  let L be up-complete lower-bounded LATTICE;
  set P = InclPoset(Ids L);
  assume
A1: SupMap L is upper_adjoint;
  for x being Element of L ex I being Ideal of L st x <= sup I & for J
  being Ideal of L st x <= sup J holds I c= J
  proof
    set r = SupMap L;
    let x be Element of L;
    set I9 = inf(r"(uparrow x));
    reconsider I = I9 as Ideal of L by YELLOW_2:41;
A2: for J being Ideal of L st x <= sup J holds I c= J
    proof
      let J be Ideal of L;
      reconsider J9= J as Element of P by YELLOW_2:41;
      assume x <= sup J;
      then x <= r.J9 by YELLOW_2:def 3;
      then J in dom r & r.J9 in uparrow x by WAYBEL_0:18,YELLOW_2:50;
      then J9 in r"(uparrow x) by FUNCT_1:def 7;
      then I9 <= J9 by YELLOW_2:22;
      hence thesis by YELLOW_1:3;
    end;
    consider d being Function of L, P such that
A3: [r,d] is Galois by A1,WAYBEL_1:def 11;
    d.x is_minimum_of r"(uparrow x) by A3,WAYBEL_1:10;
    then I in r"(uparrow x) by WAYBEL_1:def 6;
    then r.I in uparrow x by FUNCT_1:def 7;
    then x <= r.I9 by WAYBEL_0:18;
    then x <= sup I by YELLOW_2:def 3;
    hence thesis by A2;
  end;
  hence thesis by Lm4;
end;
