reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace,
  X0, X1 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X0, X1, X2 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X for non empty TopSpace,
  H, G for Subset of X;
reserve A for Subset of X;

theorem Th98:
  for X0 being non empty SubSpace of X st the carrier of X0 = A
  for x0 being Point of X0 holds (modid(X,A))|X0 is_continuous_at x0
proof
  let X0 be non empty SubSpace of X;
  assume
A1: the carrier of X0 = A;
  let x0 be Point of X0;
  now
    x0 in the carrier of X0 & the carrier of X0 c= the carrier of X by
BORSUK_1:1;
    then reconsider x = x0 as Point of X;
    let W be Subset of X modified_with_respect_to A;
    assume that
A2: W is open and
A3: (modid(X,A)|X0).x0 in W;
    consider H, G being Subset of X such that
A4: W = H \/ (G /\ A) and
A5: H in the topology of X & G in the topology of X by A2;
    reconsider H, G as Subset of X;
A6: (H /\ A) \/ (G /\ A) c= W by A4,XBOOLE_1:9,17;
    (modid(X,A)|X0).x0 = (id (the carrier of X)).x by FUNCT_1:49
      .= x;
    then x in H or x in G /\ A by A3,A4,XBOOLE_0:def 3;
    then x in H /\ A or x in G /\ A by A1,XBOOLE_0:def 4;
    then
A7: x in (H /\ A) \/ (G /\ A) by XBOOLE_0:def 3;
A8: (modid(X,A)|X0).:((H \/ G) /\ A) = (id (the carrier of X)).:((H \/ G)
    /\ A) by A1,FUNCT_2:97,XBOOLE_1:17
      .= (H \/ G) /\ A by FUNCT_1:92;
    thus ex V being Subset of X0 st V is open & x0 in V & (modid(X,A)|X0).: V
    c= W
    proof
      reconsider V = (H \/ G) /\ A as Subset of X0 by A1,XBOOLE_1:17;
      take V;
      H is open & G is open by A5;
      then
A9:   H \/ G is open;
      V = (H \/ G) /\ [#]X0 by A1;
      hence thesis by A7,A8,A6,A9,TOPS_2:24,XBOOLE_1:23;
    end;
  end;
  hence thesis by Th43;
end;
