
theorem
  for N being finite LATTICE holds N, InclPoset Ids N are_isomorphic
proof
  let N be finite LATTICE;
  set I = InclPoset Ids N;
  take i = IdsMap N;
  N is non empty & I is non empty implies i is one-to-one monotone & ex s
  being Function of I,N st s = i" & s is monotone
  proof
    assume that
    N is non empty and
    I is non empty;
    thus i is one-to-one monotone;
    take s = SupMap N;
    [i,s] is Galois by WAYBEL_1:57;
    then i is onto by WAYBEL_1:24;
    then
A1: rng i = the carrier of I by FUNCT_2:def 3;
A2: for y, x being object holds y in rng i & x = s.y iff x in dom i & y = i.x
    proof
      let y, x be object;
A3:   dom i = the carrier of N by FUNCT_2:def 1;
      hereby
        assume that
A4:     y in rng i and
A5:     x = s.y;
        reconsider Y = y as Element of I by A4;
        x = s.Y by A5;
        hence x in dom i by A3;
        reconsider Y1 = Y as Ideal of N by YELLOW_2:41;
        thus i.x = i.sup Y1 by A5,YELLOW_2:def 3
          .= downarrow sup Y1 by YELLOW_2:def 4
          .= y by WAYBEL14:5,WAYBEL_3:16;
      end;
      assume that
A6:   x in dom i and
A7:   y = i.x;
      reconsider X = x as Element of N by A6;
A8:   y = i.X by A7;
      then reconsider Y = y as Ideal of N by YELLOW_2:41;
      thus y in rng i by A1,A8;
      thus s.y = sup Y by YELLOW_2:def 3
        .= sup downarrow X by A7,YELLOW_2:def 4
        .= x by WAYBEL_0:34;
    end;
    dom s = the carrier of I by FUNCT_2:def 1;
    hence s = i" by A1,A2,FUNCT_1:32;
    thus thesis;
  end;
  hence thesis by WAYBEL_0:def 38;
end;
