
theorem Th65: :: PROPOSITION 4.3.(iii)
  for L be lower-bounded continuous sup-Semilattice for B be
  with_bottom CLbasis of L holds rng supMap subrelstr B = the carrier of L
proof
  let L be lower-bounded continuous sup-Semilattice;
  let B be with_bottom CLbasis of L;
A1: Bottom L in B by Def8;
  thus rng supMap subrelstr B = the carrier of L
  proof
    thus rng supMap subrelstr B c= the carrier of L;
    let x be object;
    assume x in the carrier of L;
    then reconsider x1 = x as Element of L;
    set z = waybelow x1 /\ B;
    z is Subset of B by XBOOLE_1:17;
    then reconsider z as Subset of subrelstr B by YELLOW_0:def 15;
A2: now
      let a,b be Element of subrelstr B;
      reconsider a1 = a, b1 = b as Element of L by YELLOW_0:58;
      assume that
A3:   a in z and
A4:   b <= a;
      a in waybelow x1 by A3,XBOOLE_0:def 4;
      then
A5:   a1 << x1 by WAYBEL_3:7;
      b1 <= a1 by A4,YELLOW_0:59;
      then b1 << x1 by A5,WAYBEL_3:2;
      then
A6:   b in waybelow x1 by WAYBEL_3:7;
      b in the carrier of subrelstr B;
      then b in B by YELLOW_0:def 15;
      hence b in z by A6,XBOOLE_0:def 4;
    end;
    Bottom L << x1 by WAYBEL_3:4;
    then
A7: Bottom L in waybelow x1 by WAYBEL_3:7;
    waybelow x1 /\ B is join-closed by Th33;
    then reconsider z as Ideal of subrelstr B by A1,A7,A2,WAYBEL10:23
,WAYBEL_0:def 19,XBOOLE_0:def 4;
    z in the set of all  X where X is Ideal of subrelstr B ;
    then z in Ids subrelstr B by WAYBEL_0:def 23;
    then
A8: z in dom supMap subrelstr B by Th51;
    x = "\/"(z,L) by Def7
      .= (supMap subrelstr B).z by Def10;
    hence thesis by A8,FUNCT_1:def 3;
  end;
end;
