
theorem Th5:
  for L1,L2,L3 be non empty Poset for g1 be Function of L1,L2 for
g2 be Function of L2,L3 for d1 be Function of L2,L1 for d2 be Function of L3,L2
  st [g1,d1] is Galois & [g2,d2] is Galois holds [g2*g1,d1*d2] is Galois
proof
  let L1,L2,L3 be non empty Poset;
  let g1 be Function of L1,L2;
  let g2 be Function of L2,L3;
  let d1 be Function of L2,L1;
  let d2 be Function of L3,L2;
  assume that
A1: [g1,d1] is Galois and
A2: [g2,d2] is Galois;
A3: d1 is monotone by A1,WAYBEL_1:8;
A4: g2 is monotone by A2,WAYBEL_1:8;
A5: now
    let s be Element of L3, t be Element of L1;
    s in the carrier of L3;
    then
A6: s in dom d2 by FUNCT_2:def 1;
    t in the carrier of L1;
    then
A7: t in dom g1 by FUNCT_2:def 1;
    thus s <= (g2*g1).t implies (d1*d2).s <= t
    proof
      assume s <= (g2*g1).t;
      then s <= g2.(g1.t) by A7,FUNCT_1:13;
      then d2.s <= g1.t by A2,WAYBEL_1:8;
      then d1.(d2.s) <= d1.(g1.t) by A3;
      then
A8:   d1.(d2.s) <= (d1*g1).t by A7,FUNCT_1:13;
      d1*g1 <= id L1 by A1,WAYBEL_1:18;
      then (d1*g1).t <= (id L1).t by YELLOW_2:9;
      then d1.(d2.s) <= (id L1).t by A8,ORDERS_2:3;
      then d1.(d2.s) <= t;
      hence thesis by A6,FUNCT_1:13;
    end;
    thus (d1*d2).s <= t implies s <= (g2*g1).t
    proof
      assume (d1*d2).s <= t;
      then d1.(d2.s) <= t by A6,FUNCT_1:13;
      then d2.s <= g1.t by A1,WAYBEL_1:8;
      then g2.(d2.s) <= g2.(g1.t) by A4;
      then
A9:   (g2*d2).s <= g2.(g1.t) by A6,FUNCT_1:13;
      id L3 <= g2*d2 by A2,WAYBEL_1:18;
      then (id L3).s <= (g2*d2).s by YELLOW_2:9;
      then (id L3).s <= g2.(g1.t) by A9,ORDERS_2:3;
      then s <= g2.(g1.t);
      hence thesis by A7,FUNCT_1:13;
    end;
  end;
  d2 is monotone by A2,WAYBEL_1:8;
  then
A10: d1*d2 is monotone by A3,YELLOW_2:12;
  g1 is monotone by A1,WAYBEL_1:8;
  then g2*g1 is monotone by A4,YELLOW_2:12;
  hence thesis by A10,A5;
end;
