
theorem Th10:
  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 g is monotone & 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, d be Function of T,S;
  hereby
    assume
A1: [g,d] is Galois;
    hence g is monotone by Th8;
    let t be Element of T;
    thus d.t is_minimum_of g"(uparrow t)
    proof
      set X = g"(uparrow t);
      t <= g.(d.t) by A1,Th8;
      then g.(d.t) in uparrow t by WAYBEL_0:18;
      then
A2:   d.t in X by FUNCT_2:38;
      then
A3:   for s being Element of S st s is_<=_than X holds d.t >= s;
A4:   d.t is_<=_than X
      proof
        let s be Element of S;
        assume s in X;
        then g.s in uparrow t by FUNCT_1:def 7;
        then t <= g.s by WAYBEL_0:18;
        hence d.t <= s by A1,Th8;
      end;
      hence ex_inf_of X,S & d.t = inf X by A3,YELLOW_0:31;
      thus thesis by A4,A2,A3,YELLOW_0:31;
    end;
  end;
  assume that
A5: g is monotone and
A6: for t being Element of T holds d.t is_minimum_of g"(uparrow t);
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;
    set X = g"(uparrow t);
    hereby
      assume t <= g.s;
      then g.s in uparrow t by WAYBEL_0:18;
      then
A8:   s in X by FUNCT_2:38;
A9:   d.t is_minimum_of g"(uparrow t) by A6;
      then ex_inf_of X,S;
      then X is_>=_than inf X by YELLOW_0:def 10;
      then s >= inf X by A8;
      hence d.t <= s by A9;
    end;
A10: d.t is_minimum_of X by A6;
    then inf X in X;
    then g.(inf X) in uparrow t by FUNCT_1:def 7;
    then g.(inf X) >= t by WAYBEL_0:18;
    then
A11: g.(d.t) >= t by A10;
    assume d.t <= s;
    then g.(d.t) <= g.s by A5;
    hence thesis by A11,ORDERS_2:3;
  end;
  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;
A13: d.t2 is_minimum_of g"(uparrow t2) by A6;
    then
A14: ex_inf_of g"(uparrow t2),S;
A15: d.t1 is_minimum_of g"(uparrow t1) by A6;
    then ex_inf_of g"(uparrow t1),S;
    then inf (g"(uparrow t1)) <= inf (g"(uparrow t2)) by A14,A12,RELAT_1:143
,YELLOW_0:35;
    then d.t1 <= inf (g"(uparrow t2)) by A15;
    hence d.t1 <= d.t2 by A13;
  end;
  hence thesis by A5,A7;
end;
