
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 one-to-one iff d is onto
proof
  let S,T be non empty Poset,g be Function of S,T, d be Function of T,S;
  assume
A1: [g,d] is Galois;
  hereby
A2: d*g <= id S & id T <= g*d by A1,Th18;
    g is monotone & d is monotone by A1,Th8;
    then
A3: g = g*d*g by A2,Th20
      .= g*(d*g) by RELAT_1:36;
A4: the carrier of S = dom g & the carrier of S = dom (d*g) by FUNCT_2:def 1;
A5: rng (d*g) c= the carrier of S;
    assume g is one-to-one;
    then d*g = id S by A4,A5,A3,FUNCT_1:28;
    hence d is onto by FUNCT_2:23;
  end;
  assume d is onto;
  then for s being Element of S holds g.s is_maximum_of d"{s} by A1,Th25;
  then d*g = id S by Th26;
  hence thesis by FUNCT_2:23;
end;
