theorem Th89:
  for A being Subset of X holds {G /\ A where G is Subset of X :
  G in the topology of X} c= A-extension_of_the_topology_of X
proof
  let A be Subset of X;
  now
    {}X = ({});
    then reconsider H = {} as Subset of X;
    let W be object;
    assume W in {G /\ A where G is Subset of X : G in the topology of X};
    then consider G being Subset of X such that
A1: W = G /\ A & G in the topology of X;
    G /\ A = H \/ (G /\ A) & H 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;
