
theorem
  for S,T being non empty Poset
  for f1 being Function of S,T, g1 being Function of T,S
  for f2 being Function of S~,T~, g2 being Function of T~,S~ st
  f1 = f2 & g1 = g2 holds [f1,g1] is Galois iff [g2,f2] is Galois
proof
  let S,T be non empty Poset;
  let f1 be Function of S,T, g1 be Function of T,S;
  let f2 be Function of S~,T~, g2 be Function of T~,S~ such that
A1: f1 = f2 and
A2: g1 = g2;
  hereby
    assume
A3: [f1,g1] is Galois;
    then f1 is monotone by WAYBEL_1:8;
    then
A4: f2 is monotone by A1,Th42;
A5: now
      let s be Element of S~, t be Element of T~;
A6:   (f1.~s)~ = f1.~s & (g1.~t)~ = g1.~t;
      ~t <= f1.~s iff g1.~t <= ~s by A3,WAYBEL_1:8;
      hence g2.t >= s iff t >= f2.s by A1,A2,A6,Th2;
    end;
    g1 is monotone by A3,WAYBEL_1:8;
    then g2 is monotone by A2,Th42;
    hence [g2,f2] is Galois by A4,A5;
  end;
  assume
A7: [g2,f2] is Galois;
  then f2 is monotone by WAYBEL_1:8;
  then
A8: f1 is monotone by A1,Th42;
A9: now
    let t be Element of T, s be Element of S;
A10: ~(f2.(s~)) = f2.(s~) & ~(g2.(t~)) = g2.(t~);
    s~ <= g2.(t~) iff f2.(s~) <= t~ by A7,WAYBEL_1:8;
    hence t <= f1.s iff g1.t <= s by A1,A2,A10,Th2;
  end;
  g2 is monotone by A7,WAYBEL_1:8;
  then g1 is monotone by A2,Th42;
  hence thesis by A8,A9;
end;
