
theorem Th8:

:: 1.2. LEMMA, (i) generalized, p. 143
  for S,T being lower complete TopLattice, f being Function of S, T
  st for X being non empty Subset of S holds f preserves_inf_of X holds f is
  continuous
proof
  let S,T be lower complete non empty TopLattice;
  reconsider BB = the set of all (uparrow x)` where x is Element of T as
  prebasis of T by Def1;
  let f be Function of S,T such that
A1: for X being non empty Subset of S holds f preserves_inf_of X;
  now
    let A be Subset of T;
A2: ex_inf_of f"A`, S by YELLOW_0:17;
A3: ex_inf_of A`, T by YELLOW_0:17;
A4: ex_inf_of f.:(f"A`), T by YELLOW_0:17;
    assume A in BB;
    then consider x being Element of T such that
A5: A = (uparrow x)`;
    set s = inf (f"uparrow x);
    per cases;
    suppose
      f"A` = {}S;
      hence f"A` is closed;
    end;
    suppose
      f"A` <> {};
      then f preserves_inf_of f"A` by A1;
      then
A6:   f.s = inf (f.:(f"A`)) by A5,A2;
      inf A` = x by A5,WAYBEL_0:39;
      then
A7:   x <= f.s by A6,A3,A4,FUNCT_1:75,YELLOW_0:35;
      f"A` = uparrow s
      proof
        thus f"A` c= uparrow s
        proof
          let a be object;
          assume
A8:       a in f"A`;
          then reconsider a as Element of S;
          s <= a by A5,A8,YELLOW_2:22;
          hence thesis by WAYBEL_0:18;
        end;
        let a be object;
        assume
A9:     a in uparrow s;
        then reconsider a as Element of S;
        f preserves_inf_of {s,a} by A1;
        then
A10:    f.(s"/\"a) = (f.s)"/\"(f.a) by YELLOW_3:8;
        s <= a by A9,WAYBEL_0:18;
        then f.s = (f.s)"/\"(f.a) by A10,YELLOW_5:10;
        then f.s <= f.a by YELLOW_0:23;
        then x <= f.a by A7,ORDERS_2:3;
        then f.a in uparrow x by WAYBEL_0:18;
        hence thesis by A5,FUNCT_2:38;
      end;
      hence f"A` is closed by Th4;
    end;
  end;
  hence thesis by YELLOW_9:35;
end;
