reserve X for non empty TopSpace,
  D for Subset of X;
reserve D for non empty set,
  d0 for Element of D;

theorem
  for D being non empty set, d0 being Element of D for A being Subset of
  STS(D,d0) st A = {d0} holds 1TopSp(D) = STS(D,d0) modified_with_respect_to A
proof
  let D be non empty set, d0 be Element of D;
  set G = {P where P is Subset of D : d0 in P & P <> D};
  set T = (bool D) \ G;
  let A be Subset of STS(D,d0);
  assume
A1: A = {d0};
A2: A-extension_of_the_topology_of STS(D,d0) = {H \/ (F /\ A) where H is
Subset of STS(D,d0), F is Subset of STS(D,d0) : H in T & F in T} by
TMAP_1:def 7;
  now
    reconsider F=D as Subset of D by Lm1;
    let W be object;
    assume W in G;
    then consider P being Subset of D such that
A3: W = P and
A4: d0 in P and
    P <> D;
    set H = P \ {d0};
    reconsider H as Subset of D;
    A c= P by A1,A4,ZFMISC_1:31;
    then
A5: W = H \/ A by A1,A3,XBOOLE_1:45;
    not ex K being Subset of D st K = F & d0 in K & K <> D;
    then not F in G;
    then
A6: F in T by XBOOLE_0:def 5;
    d0 in {d0} by TARSKI:def 1;
    then not ex K being Subset of D st K = H & d0 in K & K <> D by
XBOOLE_0:def 5;
    then not H in G;
    then
A7: H in T by XBOOLE_0:def 5;
    A = F /\ A by XBOOLE_1:28;
    hence W in A-extension_of_the_topology_of STS(D,d0) by A2,A6,A7,A5;
  end;
  then
A8: G c= A-extension_of_the_topology_of STS(D,d0) by TARSKI:def 3;
  T \/ G = (bool D) \/ G by XBOOLE_1:39;
  then
A9: bool D c= T \/ G by XBOOLE_1:7;
  T c= A-extension_of_the_topology_of STS(D,d0) by TMAP_1:88;
  then T \/ G c= A-extension_of_the_topology_of STS(D,d0) by A8,XBOOLE_1:8;
  then
A10: bool D c= A-extension_of_the_topology_of STS(D,d0) by A9,XBOOLE_1:1;
  the topology of STS(D,d0) modified_with_respect_to A = A
  -extension_of_the_topology_of STS(D,d0) by TMAP_1:93
    .= the topology of 1TopSp(D) by A10,XBOOLE_0:def 10;
  hence thesis by TMAP_1:93;
end;
