
theorem Th21:
  for T being non empty TopSpace, L being TopLattice, t being
Point of T, l being Point of L, X being Subset-Family of L st the TopStruct of
  T = the TopStruct of L & t = l & X is Basis of l holds X is Basis of t
proof
  let T be non empty TopSpace, L be TopLattice, t be Point of T, l be Point of
  L, X be Subset-Family of L;
  assume
A1: the TopStruct of T = the TopStruct of L;
  then reconsider X9 = X as Subset-Family of T;
  assume
A2: t = l;
  assume
A3: X is Basis of l;
  then
A4: X c= the topology of L by TOPS_2:64;
A5: l in Intersect X by A3,YELLOW_8:def 1;
A6: for S being Subset of L st S is open & l in S ex V being Subset of L
  st V in X & V c= S by A3,YELLOW_8:def 1;
  now
    let S be Subset of T such that
A7: S is open and
A8: t in S;
    reconsider S9 = S as Subset of L by A1;
    S in the topology of T by A7,PRE_TOPC:def 2;
    then S9 is open by A1,PRE_TOPC:def 2;
    then consider V being Subset of L such that
A9: V in X & V c= S9 by A2,A6,A8;
    reconsider V as Subset of T by A1;
    take V;
    thus V in X9 & V c= S by A9;
  end;
  hence thesis by A1,A2,A4,A5,TOPS_2:64,YELLOW_8:def 1;
end;
