
theorem Th62:
  for L be up-complete non empty Poset for S be non empty full
  SubRelStr of L holds supMap S = (SupMap L)*(idsMap S)
proof
  let L be up-complete non empty Poset;
  let S be non empty full SubRelStr of L;
A1: now
    let x be object;
    thus x in dom (supMap S) implies x in dom (idsMap S) & (idsMap S).x in dom
    (SupMap L)
    proof
      assume x in dom (supMap S);
      then x is Ideal of S by Th52;
      hence x in dom (idsMap S) by Th54;
      then (idsMap S).x in rng (idsMap S) by FUNCT_1:def 3;
      then (idsMap S).x is Ideal of L by Th55;
      hence thesis by YELLOW_2:50;
    end;
    thus x in dom (idsMap S) & (idsMap S).x in dom (SupMap L) implies x in dom
    (supMap S)
    proof
      assume that
A2:   x in dom (idsMap S) and
      (idsMap S).x in dom (SupMap L);
      x is Ideal of S by A2,Th54;
      hence thesis by Th52;
    end;
  end;
  now
    let x be object;
    assume x in dom (supMap S);
    then reconsider I = x as Ideal of S by Th52;
    consider J be Subset of L such that
A3: I = J and
A4: (idsMap S).I = downarrow J by Def11;
    reconsider J as non empty directed Subset of L by A3,YELLOW_2:7;
A5: ex_sup_of J,L by WAYBEL_0:75;
    thus (supMap S).x = "\/"(I,L) by Def10
      .= sup (downarrow J) by A3,A5,WAYBEL_0:33
      .= (SupMap L).((idsMap S).x) by A4,YELLOW_2:def 3;
  end;
  hence thesis by A1,FUNCT_1:10;
end;
