
theorem :: PROPOSITION 4.3.(vii)
  for L be lower-bounded continuous sup-Semilattice for B be
  with_bottom CLbasis of L holds CompactSublatt InclPoset(Ids subrelstr B),
  subrelstr B are_isomorphic
proof
  let L be lower-bounded continuous sup-Semilattice;
  let B be with_bottom CLbasis of L;
  deffunc F(Element of subrelstr B) = downarrow $1;
A1: for x be Element of subrelstr B holds F(x) is Element of CompactSublatt
  InclPoset Ids subrelstr B
  proof
    let x be Element of subrelstr B;
    downarrow x in the set of all
 downarrow b where b is Element of subrelstr B ;
    hence thesis by Th68;
  end;
  consider f be Function of subrelstr B,CompactSublatt InclPoset Ids subrelstr
  B such that
A2: for x be Element of subrelstr B holds f.x = F(x) from FUNCT_2:sch 9
  ( A1);
  f is isomorphic by A2,WAYBEL13:13;
  then
  subrelstr B,CompactSublatt InclPoset(Ids subrelstr B) are_isomorphic by
WAYBEL_1:def 8;
  hence thesis by WAYBEL_1:6;
end;
