
theorem
  for T being non empty TopSpace, A being Subset of T, x being Point of
T holds x in Der A iff for B being Basis of x, U being Subset of T st U in B 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 B be Basis of x, U be Subset of T;
    assume U in B;
    then U is open & x in U by YELLOW_8:12;
    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 B being Basis of x, U being Subset of T st U in B 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
    set B = the Basis of x;
    let G be Subset of T;
    assume
A5: G is open;
    assume x in G;
    then consider V being Subset of T such that
A6: V in B & V c= G by A5,YELLOW_8:13;
    (ex y being Point of T st y in A /\ V & x <> y )& A /\ V c= A /\ G by A4,A6
,XBOOLE_1:26;
    then consider y being Point of T such that
A7: y in A /\ G and
A8: x <> y;
    y in A by A7,XBOOLE_0:def 4;
    then
A9: y in A \ {x} by A8,ZFMISC_1:56;
    y in G by A7,XBOOLE_0:def 4;
    hence thesis by A9,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;
