
theorem Th11:
  for S,T being non empty Poset,g being Function of S,T, d being
  Function of T,S holds [g,d] is Galois iff d is monotone & for s being Element
  of S holds g.s is_maximum_of d"(downarrow s)
proof
  let S,T be non empty Poset,g be Function of S,T, d be Function of T,S;
  hereby
    assume
A1: [g,d] is Galois;
    hence d is monotone by Th8;
    let s be Element of S;
    thus g.s is_maximum_of d"(downarrow s)
    proof
      set X = d"(downarrow s);
      s >= d.(g.s) by A1,Th8;
      then d.(g.s) in downarrow s by WAYBEL_0:17;
      then
A2:   g.s in X by FUNCT_2:38;
      then
A3:   for t being Element of T st t is_>=_than X holds g.s <= t;
A4:   g.s is_>=_than X
      proof
        let t be Element of T;
        assume t in X;
        then d.t in downarrow s by FUNCT_1:def 7;
        then s >= d.t by WAYBEL_0:17;
        hence thesis by A1,Th8;
      end;
      hence ex_sup_of X,T & g.s = sup X by A3,YELLOW_0:30;
      thus thesis by A4,A2,A3,YELLOW_0:30;
    end;
  end;
  assume that
A5: d is monotone and
A6: for s being Element of S holds g.s is_maximum_of d"(downarrow s);
A7: for t being Element of T, s being Element of S holds s >= d.t iff g.s >= t
  proof
    let t be Element of T, s be Element of S;
    set X = d"(downarrow s);
A8: g.s is_maximum_of X by A6;
    then sup X in X;
    then d.(sup X) in downarrow s by FUNCT_1:def 7;
    then d.(sup X) <= s by WAYBEL_0:17;
    then
A9: d.(g.s) <= s by A8;
    hereby
      assume s >= d.t;
      then d.t in downarrow s by WAYBEL_0:17;
      then
A10:  t in X by FUNCT_2:38;
      ex_sup_of X,T by A8;
      then X is_<=_than sup X by YELLOW_0:def 9;
      then t <= sup X by A10;
      hence g.s >= t by A8;
    end;
    assume g.s >= t;
    then d.(g.s) >= d.t by A5;
    hence thesis by A9,ORDERS_2:3;
  end;
  g is monotone
  proof
    let s1,s2 be Element of S;
    assume s1 <= s2;
    then
A11: downarrow s1 c= downarrow s2 by WAYBEL_0:21;
A12: g.s2 is_maximum_of d"(downarrow s2) by A6;
    then
A13: ex_sup_of d"(downarrow s2),T;
A14: g.s1 is_maximum_of d"(downarrow s1) by A6;
    then ex_sup_of d"(downarrow s1),T;
    then sup (d"(downarrow s1)) <= sup (d"(downarrow s2)) by A13,A11,
RELAT_1:143,YELLOW_0:34;
    then g.s1 <= sup (d"(downarrow s2)) by A14;
    hence g.s1 <= g.s2 by A12;
  end;
  hence thesis by A5,A7;
end;
