
theorem Th9: :: 1.3.1. Characterization of Fr via basis
  for T being non empty TopSpace, A being Subset of T, p being
  Point of T holds p in Fr A iff for U being Subset of T st U is open & p in U
  holds A meets U & U \ A <> {}
proof
  let T be non empty TopSpace, A be Subset of T, p be Point of T;
  hereby
    assume
A1: p in Fr A;
    let U be Subset of T;
    assume
A2: U is open & p in U;
    then U meets A` by A1,TOPS_1:28;
    hence A meets U & U \ A <> {} by A1,A2,Th1,TOPS_1:28;
  end;
  assume
A3: for U being Subset of T st U is open & p in U holds A meets U & U \
  A <> {};
  for U being Subset of T st U is open & p in U holds A meets U & U meets A`
  proof
    let U be Subset of T;
    assume
A4: U is open & p in U;
    then U \ A <> {} by A3;
    hence thesis by A3,A4,Th1;
  end;
  hence thesis by TOPS_1:28;
end;
