
theorem Th15:
  for S,T being non empty Poset, d being Function of T,S st T is
complete & d is sups-preserving ex g being Function of S,T st [g,d] is Galois &
  for s being Element of S holds g.s is_maximum_of d"(downarrow s)
proof
  let S,T be non empty Poset, d be Function of T,S;
  assume that
A1: T is complete and
A2: d is sups-preserving;
  defpred P[object,object] means
ex s being Element of S st s = $1 & $2 = sup (d"(
  downarrow s));
A3: for e being object st e in the carrier of S
    ex u being object st u in the carrier of T & P[e,u]
  proof
    let e be object;
    assume e in the carrier of S;
    then reconsider s = e as Element of S;
    take sup (d"(downarrow s));
    thus thesis;
  end;
  consider g being Function of the carrier of S, the carrier of T such that
A4: for e being object st e in the carrier of S holds P[e,g.e] from FUNCT_2
  :sch 1(A3);
A5: for s being Element of S holds g.s = sup (d"(downarrow s))
  proof
    let s be Element of S;
    ex s1 being Element of S st s1 = s & g.s = sup (d"(downarrow s1)) by A4;
    hence thesis;
  end;
  reconsider g as Function of S,T;
  for X being Ideal of T holds d preserves_sup_of X by A2;
  then
A6: d is monotone by WAYBEL_0:72;
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;
A8: ex_sup_of downarrow s,S by WAYBEL_0:34;
A9: ex_sup_of d"(downarrow s),T by A1,YELLOW_0:17;
    then sup (d"(downarrow s)) is_>=_than d"(downarrow s) by YELLOW_0:30;
    then
A10: g.s is_>=_than d"(downarrow s) by A5;
    hereby
      assume s >= d.t;
      then d.t in downarrow s by WAYBEL_0:17;
      then t in d"(downarrow s) by FUNCT_2:38;
      hence g.s >= t by A10;
    end;
    d preserves_sup_of (d"(downarrow s)) by A2;
    then
    ex_sup_of d.:(d"(downarrow s)),S & d.(sup (d"(downarrow s))) = sup (d
    .:(d"( downarrow s))) by A9;
    then d.(sup (d"(downarrow s))) <= sup(downarrow s) by A8,FUNCT_1:75
,YELLOW_0:34;
    then
A11: d.(sup (d"(downarrow s))) <= s by WAYBEL_0:34;
    assume g.s >= t;
    then d.(g.s) >= d.t by A6;
    then d.(sup (d"(downarrow s))) >= d.t by A5;
    hence thesis by A11,ORDERS_2:3;
  end;
  take g;
  g is monotone
  proof
    let s1,s2 be Element of S;
    assume s1 <= s2;
    then
A12: downarrow s1 c= downarrow s2 by WAYBEL_0:21;
    ex_sup_of d"(downarrow s1),T & ex_sup_of d"(downarrow s2),T by A1,
YELLOW_0:17;
    then sup (d"(downarrow s1)) <= sup (d"(downarrow s2)) by A12,RELAT_1:143
,YELLOW_0:34;
    then g.s1 <= sup (d"(downarrow s2)) by A5;
    hence g.s1 <= g.s2 by A5;
  end;
  hence [g,d] is Galois by A6,A7;
  let s be Element of S;
  thus
A13: ex_sup_of d"(downarrow s),T by A1,YELLOW_0:17;
  thus
A14: g.s = sup (d"(downarrow s)) by A5;
A15: ex_sup_of downarrow s,S by WAYBEL_0:34;
  d preserves_sup_of (d"(downarrow s)) by A2;
  then
  ex_sup_of d.:(d"(downarrow s)),S & d.(sup (d"(downarrow s))) = sup (d.:
  (d"( downarrow s))) by A13;
  then d.(sup (d"(downarrow s))) <= sup(downarrow s) by A15,FUNCT_1:75
,YELLOW_0:34;
  then d.(sup (d"(downarrow s))) <= s by WAYBEL_0:34;
  then d.(g.s) <= s by A5;
  then d.(g.s) in downarrow s by WAYBEL_0:17;
  hence thesis by A14,FUNCT_2:38;
end;
