
theorem
  for S,T being non empty Poset,g being Function of S,T, d being
  Function of T,S st [g,d] is Galois holds g is onto iff d is one-to-one
proof
  let S,T be non empty Poset,g be Function of S,T, d be Function of T,S;
A1: the carrier of T = dom d & the carrier of T = dom (g*d) by FUNCT_2:def 1;
  assume
A2: [g,d] is Galois;
  then
A3: d*g <= id S & id T <= g*d by Th18;
  hereby
    assume g is onto;
    then for t being Element of T holds d.t is_minimum_of g"{t} by A2,Th22;
    then g*d = id T by Th23;
    hence d is one-to-one by FUNCT_2:23;
  end;
A4: rng (g*d) c= the carrier of T;
  g is monotone & d is monotone by A2,Th8;
  then
A5: d = d*g*d by A3,Th20
    .= d*(g*d) by RELAT_1:36;
  assume d is one-to-one;
  then g*d = id T by A1,A4,A5,FUNCT_1:28;
  hence thesis by FUNCT_2:23;
end;
