
theorem :: PROPOSITION 4.4. (1) iff (3)
  for L be continuous lower-bounded LATTICE holds L is algebraic iff ex
B be with_bottom CLbasis of L st for B1 be with_bottom CLbasis of L holds B c=
  B1
proof
  let L be continuous lower-bounded LATTICE;
  thus L is algebraic implies ex B be with_bottom CLbasis of L st for B1 be
  with_bottom CLbasis of L holds B c= B1
  proof
    assume
A1: L is algebraic;
    then
A2: for B be with_bottom CLbasis of L holds the carrier of CompactSublatt
    L c= B by Lm4;
    the carrier of CompactSublatt L is with_bottom CLbasis of L by A1,Lm4;
    hence thesis by A2;
  end;
  thus thesis by Lm5;
end;
