
theorem Th25:
  for S,T being non empty Poset,g being Function of S,T, d being
Function of T,S st [g,d] is Galois & d is onto for s being Element of S holds g
  .s is_maximum_of d"{s}
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: d is onto;
A3: d is monotone by A1,Th8;
  let s be Element of S;
A4: rng d = the carrier of S by A2,FUNCT_2:def 3;
  then
A5: d.:(d"(downarrow s)) = downarrow s by FUNCT_1:77;
A6: g.s is_maximum_of (d"(downarrow s)) by A1,Th11;
  then
A7: g.s = sup (d"(downarrow s));
  g.s in d"(downarrow s) by A6;
  then d.(g.s) in d.:(d"(downarrow s)) by FUNCT_2:35;
  then
A8: s >= d.(g.s) by A5,WAYBEL_0:17;
  ex_sup_of d"(downarrow s),T by A6;
  then
A9: g.s is_>=_than d"(downarrow s) by A7,YELLOW_0:30;
  consider t being object such that
A10: t in the carrier of T and
A11: d.t = s by A4,FUNCT_2:11;
  reconsider t as Element of T by A10;
A12: s in {s} by TARSKI:def 1;
A13: {s} c= downarrow {s} by WAYBEL_0:16;
  then t in d"(downarrow s) by A11,A12,FUNCT_2:38;
  then g.s >= t by A9;
  then d.(g.s) >= s by A11,A3;
  then
A14: d.(g.s) = s by A8,ORDERS_2:2;
  then
A15: g.s in d"{s} by A12,FUNCT_2:38;
A16: d"{s} c= d"(downarrow s) by RELAT_1:143,WAYBEL_0:16;
  thus
A17: ex_sup_of d"{s},T
  proof
    take g.s;
    thus d"{s} is_<=_than g.s by A9,A16;
    thus for b be Element of T st d"{s} is_<=_than b holds b >= g.s by A15;
    let c be Element of T;
    assume d"{s} is_<=_than c;
    then
A18: c >= g.s by A15;
    assume for b being Element of T st d"{s} is_<=_than b holds b >= c;
    then g.s >= c by A9,A16,YELLOW_0:9;
    hence thesis by A18,ORDERS_2:2;
  end;
  then sup (d"{s}) is_>=_than d"{s} by YELLOW_0:30;
  then
A19: sup (d"{s}) >= g.s by A15;
  ex_sup_of d"(downarrow s),T by A6;
  then sup (d"{s}) <= g.s by A7,A13,A17,RELAT_1:143,YELLOW_0:34;
  hence g.s = sup(d"{s}) by A19,ORDERS_2:2;
  hence thesis by A12,A14,FUNCT_2:38;
end;
