
theorem Th14:
  for S,T being non empty Poset,g being Function of S,T st S is
complete & g is infs-preserving ex d being Function of T,S st [g,d] is Galois &
  for t being Element of T holds d.t is_minimum_of g"(uparrow t)
proof
  let S,T be non empty Poset,g be Function of S,T;
  assume that
A1: S is complete and
A2: g is infs-preserving;
  defpred P[object,object] means
ex t being Element of T st t = $1 & $2 = inf (g"(
  uparrow t));
A3: for e being object st e in the carrier of T
   ex u being object st u in the carrier of S & P[e,u]
  proof
    let e be object;
    assume e in the carrier of T;
    then reconsider t = e as Element of T;
    take inf (g"(uparrow t));
    thus thesis;
  end;
  consider d being Function of the carrier of T, the carrier of S such that
A4: for e being object st e in the carrier of T holds P[e,d.e] from FUNCT_2
  :sch 1(A3);
A5: for t being Element of T holds d.t = inf (g"(uparrow t))
  proof
    let t be Element of T;
    ex t1 being Element of T st t1 = t & d.t = inf (g"(uparrow t1)) by A4;
    hence thesis;
  end;
  reconsider d as Function of T,S;
  for X being Filter of S holds g preserves_inf_of X by A2;
  then
A6: g is monotone by WAYBEL_0:69;
A7: for t being Element of T, s being Element of S holds t <= g.s iff d.t <= s
  proof
    let t be Element of T, s be Element of S;
A8: ex_inf_of uparrow t,T by WAYBEL_0:39;
A9: ex_inf_of g"(uparrow t),S by A1,YELLOW_0:17;
    then inf (g"(uparrow t)) is_<=_than g"(uparrow t) by YELLOW_0:31;
    then
A10: d.t is_<=_than g"(uparrow t) by A5;
    hereby
      assume t <= g.s;
      then g.s in uparrow t by WAYBEL_0:18;
      then s in g"(uparrow t) by FUNCT_2:38;
      hence d.t <= s by A10;
    end;
    g preserves_inf_of (g"(uparrow t)) by A2;
    then
    ex_inf_of g.:(g"(uparrow t)),T & g.(inf (g"(uparrow t))) = inf (g.:(g
    "( uparrow t))) by A9;
    then g.(inf (g"(uparrow t))) >= inf(uparrow t) by A8,FUNCT_1:75,YELLOW_0:35
;
    then
A11: g.(inf (g"(uparrow t))) >= t by WAYBEL_0:39;
    assume d.t <= s;
    then g.(d.t) <= g.s by A6;
    then g.(inf (g"(uparrow t))) <= g.s by A5;
    hence thesis by A11,ORDERS_2:3;
  end;
  take d;
  d is monotone
  proof
    let t1,t2 be Element of T;
    assume t1 <= t2;
    then
A12: uparrow t2 c= uparrow t1 by WAYBEL_0:22;
    ex_inf_of g"(uparrow t1),S & ex_inf_of g"(uparrow t2),S by A1,YELLOW_0:17;
    then inf (g"(uparrow t1)) <= inf (g"(uparrow t2)) by A12,RELAT_1:143
,YELLOW_0:35;
    then d.t1 <= inf (g"(uparrow t2)) by A5;
    hence d.t1 <= d.t2 by A5;
  end;
  hence [g,d] is Galois by A6,A7;
  let t be Element of T;
  thus
A13: ex_inf_of g"(uparrow t),S by A1,YELLOW_0:17;
  thus
A14: d.t = inf (g"(uparrow t)) by A5;
A15: ex_inf_of uparrow t,T by WAYBEL_0:39;
  g preserves_inf_of (g"(uparrow t)) by A2;
  then
  ex_inf_of g.:(g"(uparrow t)),T & g.(inf (g"(uparrow t))) = inf (g.:(g"(
  uparrow t))) by A13;
  then g.(inf (g"(uparrow t))) >= inf(uparrow t) by A15,FUNCT_1:75,YELLOW_0:35;
  then g.(inf (g"(uparrow t))) >= t by WAYBEL_0:39;
  then g.(d.t) >= t by A5;
  then g.(d.t) in uparrow t by WAYBEL_0:18;
  hence thesis by A14,FUNCT_2:38;
end;
