
theorem Th22:
  for S,T being non empty Poset,g being Function of S,T, d being
Function of T,S st [g,d] is Galois & g is onto for t being Element of T holds d
  .t is_minimum_of g"{t}
proof
  let S,T be non empty Poset,g be Function of S,T, d be Function of T,S;
  assume that
A1: [g,d] is Galois and
A2: g is onto;
A3: g is monotone by A1,Th8;
  let t be Element of T;
A4: rng g = the carrier of T by A2,FUNCT_2:def 3;
  then
A5: g.:(g"(uparrow t)) = uparrow t by FUNCT_1:77;
A6: d.t is_minimum_of g"(uparrow t) by A1,Th10;
  then
A7: d.t = inf (g"(uparrow t));
  d.t in g"(uparrow t) by A6;
  then g.(d.t) in g.:(g"(uparrow t)) by FUNCT_2:35;
  then
A8: t <= g.(d.t) by A5,WAYBEL_0:18;
  ex_inf_of g"(uparrow t),S by A6;
  then
A9: d.t is_<=_than g"(uparrow t) by A7,YELLOW_0:31;
  consider s being object such that
A10: s in the carrier of S and
A11: g.s = t by A4,FUNCT_2:11;
  reconsider s as Element of S by A10;
A12: t in {t} by TARSKI:def 1;
A13: {t} c= uparrow {t} by WAYBEL_0:16;
  then s in g"(uparrow t) by A11,A12,FUNCT_2:38;
  then d.t <= s by A9;
  then g.(d.t) <= t by A11,A3;
  then
A14: g.(d.t) = t by A8,ORDERS_2:2;
  then
A15: d.t in g"{t} by A12,FUNCT_2:38;
A16: g"{t} c= g"(uparrow t) by RELAT_1:143,WAYBEL_0:16;
  thus
A17: ex_inf_of g"{t},S
  proof
    take d.t;
    thus g"{t} is_>=_than d.t by A9,A16;
    thus for b be Element of S st g"{t} is_>=_than b holds b <= d.t by A15;
    let c be Element of S;
    assume g"{t} is_>=_than c;
    then
A18: c <= d.t by A15;
    assume for b being Element of S st g"{t} is_>=_than b holds b <= c;
    then d.t <= c by A9,A16,YELLOW_0:9;
    hence thesis by A18,ORDERS_2:2;
  end;
  then inf (g"{t}) is_<=_than g"{t} by YELLOW_0:31;
  then
A19: inf (g"{t}) <= d.t by A15;
  ex_inf_of g"(uparrow t),S by A6;
  then inf (g"{t}) >= d.t by A7,A13,A17,RELAT_1:143,YELLOW_0:35;
  hence d.t = inf(g"{t}) by A19,ORDERS_2:2;
  hence thesis by A12,A14,FUNCT_2:38;
end;
