reserve a,b,c for set;

theorem
  for T being non empty TopSpace for A being Subset of T for x being
Element of T holds x in Cl A iff for B being Basis of x for U being set st U in
  B holds U meets A
proof
  let T be non empty TopSpace;
  let A be Subset of T;
  let x be Element of T;
  set B = the Basis of x;
  thus x in Cl A implies for B being Basis of x for U being set st U in B
  holds U meets A by Lm2;
  assume for B being Basis of x for U being set st U in B holds U meets A;
  then for U being set st U in B holds U meets A;
  hence thesis by Lm3;
end;
