
theorem Th6:
  for L1,L2 be non empty Poset for f be Function of L1,L2 for f1 be
  Function of L2,L1 st f1 = (f qua Function)" & f is isomorphic holds [f,f1] is
  Galois & [f1,f] is Galois
proof
  let L1,L2 be non empty Poset;
  let f be Function of L1,L2;
  let f1 be Function of L2,L1;
  assume that
A1: f1 = (f qua Function)" and
A2: f is isomorphic;
A3: f1 is isomorphic by A1,A2,WAYBEL_0:68;
  now
    let t be Element of L2, s be Element of L1;
    s in the carrier of L1;
    then
A4: s in dom f by FUNCT_2:def 1;
A5: f1*f = id dom f by A1,A2,FUNCT_1:39
      .= id L1 by FUNCT_2:def 1;
    thus t <= f.s implies f1.t <= s
    proof
      assume t <= f.s;
      then f1.t <= f1.(f.s) by A3,WAYBEL_1:def 2;
      then f1.t <= (f1*f).s by A4,FUNCT_1:13;
      hence thesis by A5;
    end;
    t in the carrier of L2;
    then
A6: t in dom f1 by FUNCT_2:def 1;
A7: f*f1 = id rng f by A1,A2,FUNCT_1:39
      .= id L2 by A2,WAYBEL_0:66;
    thus f1.t <= s implies t <= f.s
    proof
      assume f1.t <= s;
      then f.(f1.t) <= f.s by A2,WAYBEL_1:def 2;
      then (f*f1).t <= f.s by A6,FUNCT_1:13;
      hence thesis by A7;
    end;
  end;
  hence [f,f1] is Galois by A2,A3;
  now
    let t be Element of L1, s be Element of L2;
    s in the carrier of L2;
    then
A8: s in dom f1 by FUNCT_2:def 1;
A9: f*f1 = id rng f by A1,A2,FUNCT_1:39
      .= id L2 by A2,WAYBEL_0:66;
    thus t <= f1.s implies f.t <= s
    proof
      assume t <= f1.s;
      then f.t <= f.(f1.s) by A2,WAYBEL_1:def 2;
      then f.t <= (f*f1).s by A8,FUNCT_1:13;
      hence thesis by A9;
    end;
    t in the carrier of L1;
    then
A10: t in dom f by FUNCT_2:def 1;
A11: f1*f = id dom f by A1,A2,FUNCT_1:39
      .= id L1 by FUNCT_2:def 1;
    thus f.t <= s implies t <= f1.s
    proof
      assume f.t <= s;
      then f1.(f.t) <= f1.s by A3,WAYBEL_1:def 2;
      then (f1*f).t <= f1.s by A10,FUNCT_1:13;
      hence thesis by A11;
    end;
  end;
  hence thesis by A2,A3;
end;
