
theorem Th4:
  for S,T be non empty Poset for g be Function of S,T for d be
  Function of T,S holds g is onto & [g,d] is Galois implies T,Image d
  are_isomorphic
proof
  let S,T be non empty Poset;
  let g be Function of S,T;
  let d be Function of T,S;
  assume that
A1: g is onto and
A2: [g,d] is Galois;
  d is one-to-one by A1,A2,WAYBEL_1:24;
  then
A3: (the carrier of Image d)|`d is one-to-one by FUNCT_1:58;
A4: d is monotone by A2,WAYBEL_1:8;
A5: now
    let x,y be Element of T;
    thus x <= y implies (corestr(d)).x <= (corestr(d)).y by A4,WAYBEL_1:def 2;
    thus (corestr(d)).x <= (corestr(d)).y implies x <= y
    proof
      for t be Element of T holds d.t is_minimum_of g"{t} by A1,A2,WAYBEL_1:22;
      then
A6:   g*d = id T by WAYBEL_1:23;
      assume
A7:   (corestr(d)).x <= (corestr(d)).y;
      y in the carrier of T;
      then
A8:   y in dom d by FUNCT_2:def 1;
A9:   g is monotone by A2,WAYBEL_1:8;
      d.x = (corestr(d)).x & d.y = (corestr(d)).y by WAYBEL_1:30;
      then d.x <= d.y by A7,YELLOW_0:59;
      then
A10:  g.(d.x) <= g.(d.y) by A9;
      x in the carrier of T;
      then x in dom d by FUNCT_2:def 1;
      then (g*d).x <= g.(d.y) by A10,FUNCT_1:13;
      then (g*d).x <= (g*d).y by A8,FUNCT_1:13;
      then (id T).x <= y by A6;
      hence thesis;
    end;
  end;
  rng corestr(d) = the carrier of Image d by FUNCT_2:def 3;
  then corestr(d) is isomorphic by A3,A5,WAYBEL_0:66;
  hence thesis;
end;
