
theorem
  for L being complete non empty Poset holds (IdsMap L)*(SupMap L) is
  closure & Image ((IdsMap L)*(SupMap L)),L are_isomorphic
proof
  let L be complete non empty Poset;
  set i = (IdsMap L)*(SupMap L);
A1: now
    let I be Ideal of L;
    I is Element of InclPoset(Ids L) by YELLOW_2:41;
    hence i.I = (IdsMap L).((SupMap L).I) by FUNCT_2:15
      .= (IdsMap L).(sup I) by YELLOW_2:def 3
      .= downarrow (sup I) by YELLOW_2:def 4;
  end;
  i is monotone & [IdsMap L,SupMap L] is Galois by Th57,YELLOW_2:12;
  hence i is closure by Th38;
  take f = (SupMap L)*(inclusion i);
A2: now
    let x be Element of Image i;
    let I be Ideal of L;
    assume
A3: x = I;
    hence f.I = (SupMap L).((inclusion i).I) by FUNCT_2:15
      .= (SupMap L).I by A3
      .= sup I by YELLOW_2:def 3;
  end;
A4: f is monotone by YELLOW_2:12;
A5: now
    let x,y be Element of Image i;
    consider Ix being Element of InclPoset(Ids L) such that
A6: i.Ix = x by YELLOW_2:10;
    thus x <= y implies f.x <= f.y by A4;
    assume
A7: f.x <= f.y;
    x is Element of InclPoset(Ids L) & y is Element of InclPoset(Ids L)
    by YELLOW_0:58;
    then reconsider x9=x, y9=y as Ideal of L by YELLOW_2:41;
    consider Iy being Element of InclPoset(Ids L) such that
A8: i.Iy = y by YELLOW_2:10;
    reconsider Ix,Iy as Ideal of L by YELLOW_2:41;
    reconsider i1 = downarrow (sup Ix), i2 = downarrow (sup Iy) as Element of
    InclPoset(Ids L) by YELLOW_2:41;
A9: i.Ix = downarrow (sup Ix) & i.Iy = downarrow (sup Iy) by A1;
A10: f.x9 = sup x9 & f.y9 = sup y9 by A2;
    sup downarrow (sup Ix) = sup Ix & sup downarrow (sup Iy) = sup Iy by
WAYBEL_0:34;
    then downarrow (sup Ix) c= downarrow (sup Iy) by A7,A6,A8,A9,A10,
WAYBEL_0:21;
    then i1 <= i2 by YELLOW_1:3;
    hence x <= y by A6,A8,A9,YELLOW_0:60;
  end;
A11: rng f = the carrier of L
  proof
    thus rng f c= the carrier of L;
    let x be object;
    assume x in the carrier of L;
    then reconsider x as Element of L;
A12: (SupMap L).(downarrow x) = sup downarrow x by YELLOW_2:def 3
      .= x by WAYBEL_0:34;
A13: downarrow x is Element of InclPoset(Ids L) by YELLOW_2:41;
    then i.(downarrow x) = (IdsMap L).((SupMap L).(downarrow x)) by FUNCT_2:15
      .= downarrow x by A12,YELLOW_2:def 4;
    then downarrow x in rng i by A13,FUNCT_2:4;
    then
A14: downarrow x in the carrier of Image i by YELLOW_0:def 15;
    then f.(downarrow x) = (SupMap L).((inclusion i).(downarrow x)) by
FUNCT_2:15
      .= (SupMap L).(downarrow x) by A14,FUNCT_1:18;
    hence thesis by A12,A14,FUNCT_2:4;
  end;
  f is one-to-one
  proof
    let x,y be Element of Image i;
    assume
A15: f.x = f.y;
    consider Ix being Element of InclPoset(Ids L) such that
A16: i.Ix = x by YELLOW_2:10;
    consider Iy being Element of InclPoset(Ids L) such that
A17: i.Iy = y by YELLOW_2:10;
    x is Element of InclPoset(Ids L) & y is Element of InclPoset(Ids L)
    by YELLOW_0:58;
    then reconsider x,y as Ideal of L by YELLOW_2:41;
    reconsider Ix,Iy as Ideal of L by YELLOW_2:41;
A18: sup downarrow (sup Ix) = sup Ix by WAYBEL_0:34;
A19: i.Ix = downarrow (sup Ix) & i.Iy = downarrow (sup Iy) by A1;
    f.x = sup x & f.y = sup y by A2;
    hence thesis by A15,A16,A17,A19,A18,WAYBEL_0:34;
  end;
  hence thesis by A11,A5,WAYBEL_0:66;
end;
