
theorem Th17:
  for T being non empty TopSpace, A being Subset of T, x being
  Point of T holds x in Der A iff for U being open Subset of T st x in U ex y
  being Point of T st y in A /\ U & x <> y
proof
  let T be non empty TopSpace, A be Subset of T, x be Point of T;
  hereby
    assume x in Der A;
    then x is_an_accumulation_point_of A by Th16;
    then
A1: x in Cl (A \ {x});
    let U be open Subset of T;
    assume x in U;
    then A \ {x} meets U by A1,PRE_TOPC:24;
    then consider y being object such that
A2: y in A \ {x} and
A3: y in U by XBOOLE_0:3;
    reconsider y as Point of T by A2;
    take y;
    y in A by A2,ZFMISC_1:56;
    hence y in A /\ U & x <> y by A2,A3,XBOOLE_0:def 4,ZFMISC_1:56;
  end;
  assume
A4: for U being open Subset of T st x in U ex y being Point of T st y in
  A /\ U & x <> y;
  for G being Subset of T st G is open holds x in G implies A \ {x} meets G
  proof
    let G be Subset of T;
    assume
A5: G is open;
    assume x in G;
    then consider y being Point of T such that
A6: y in A /\ G and
A7: x <> y by A4,A5;
    y in A by A6,XBOOLE_0:def 4;
    then
A8: y in A \ {x} by A7,ZFMISC_1:56;
    y in G by A6,XBOOLE_0:def 4;
    hence thesis by A8,XBOOLE_0:3;
  end;
  then x in Cl (A \ {x}) by PRE_TOPC:24;
  then x is_an_accumulation_point_of A;
  hence thesis by Th16;
end;
