theorem Th88:
  for A being Subset of X holds the topology of X c= A
  -extension_of_the_topology_of X
proof
  let A be Subset of X;
  now
    {}X = ({});
    then reconsider G = {} as Subset of X;
    let W be object;
    assume
A1: W in the topology of X;
    then reconsider H = W as Subset of X;
    H = H \/ (G /\ A) & G in the topology of X by PRE_TOPC:1;
    hence W in A-extension_of_the_topology_of X by A1;
  end;
  hence thesis by TARSKI:def 3;
end;
