
theorem Th62:
  for S being Semilattice holds (for x being Element of S holds x
"/\" is lower_adjoint) iff for x,t being Element of S holds ex_max_of {s where
  s is Element of S: x"/\"s <= t},S
proof
  let S be Semilattice;
  hereby
    assume
A1: for x being Element of S holds x "/\" is lower_adjoint;
    let x,t be Element of S;
    (x "/\") is lower_adjoint by A1;
    then consider g being Function of S,S such that
A2: [g, x "/\"] is Galois;
    set X = {s where s is Element of S: x"/\"s <= t};
A3: X = (x "/\")"(downarrow t) by Th59;
    g.t is_maximum_of (x "/\")"(downarrow t) by A2,Th11;
    then ex_sup_of X,S & "\/"(X,S)in X by A3;
    hence ex_max_of X,S;
  end;
  assume
A4: for x,t being Element of S holds ex_max_of {s where s is Element of
S: x"/\"s <= t},S;
  let x be Element of S;
  deffunc F(Element of S) = "\/"((x "/\")"(downarrow $1),S);
  consider g being Function of S,S such that
A5: for s being Element of S holds g.s = F(s) from FUNCT_2:sch 4;
  now
    let t be Element of S;
    set X = {s where s is Element of S: x"/\"s <= t};
    ex_max_of X,S by A4;
    then
A6: ex_sup_of X,S & "\/"(X,S) in X;
    X = (x "/\")"(downarrow t) & g.t = "\/"((x "/\")"(downarrow t),S) by A5
,Th59;
    hence g.t is_maximum_of (x "/\")"(downarrow t) by A6;
  end;
  then [g, x "/\"] is Galois by Th11;
  hence thesis;
end;
