
theorem Th8:
  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 & d is monotone & for t
  being Element of T, s being Element of S holds t <= g.s iff d.t <= s
proof
  let S,T be non empty Poset,g be Function of S,T, d be Function of T,S;
  hereby
    assume [g,d] is Galois;
    then consider g9 being Function of S,T, d9 being Function of T,S such that
A1: [g,d] = [g9,d9] and
A2: g9 is monotone & d9 is monotone & for t being Element of T, s
    being Element of S holds t <= g9.s iff d9.t <= s;
    g = g9 & d = d9 by A1,XTUPLE_0:1;
    hence g is monotone & d is monotone & for t being Element of T, s being
    Element of S holds t <= g.s iff d.t <= s by A2;
  end;
  thus thesis;
end;
