theorem Th96:
  for x being Point of X st not x in A holds modid(X,A) is_continuous_at x
proof
  let x be Point of X;
  assume
A1: not x in A;
  now
    let W be Subset of X modified_with_respect_to A;
    assume that
A2: W is open and
A3: (modid(X,A)).x in W;
    consider H, G being Subset of X such that
A4: W = H \/ (G /\ A) and
A5: H in the topology of X and
    G in the topology of X by A2;
A6: G /\ A c= A by XBOOLE_1:17;
A7: x in H or x in G /\ A by A3,A4,XBOOLE_0:def 3;
    thus ex V being Subset of X st V is open & x in V & (modid(X,A)).:V c= W
    proof
      reconsider H as Subset of X;
      take H;
      (modid(X,A)).:H = H by FUNCT_1:92;
      hence thesis by A1,A4,A5,A7,A6,XBOOLE_1:7;
    end;
  end;
  hence thesis by Th43;
end;
