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;
reserve X0 for non empty SubSpace of X;

theorem Th106:
  for X0, X1 being non empty SubSpace of X st X0 misses X1 for x1
  being Point of X1 holds (modid(X,X0))|X1 is_continuous_at x1
proof
  let X0, X1 be non empty SubSpace of X;
  reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
  reconsider f = (modid(X,A))|X1 as Function of X1,X modified_with_respect_to
  X0 by Def10;
  assume
A1: X0 misses X1;
  let x1 be Point of X1;
  (the carrier of X1) misses A by A1,TSEP_1:def 3;
  then
A2: (modid(X,A))|X1 is_continuous_at x1 by Th97;
  now
    let W be Subset of X modified_with_respect_to X0;
    reconsider W0 = W as Subset of (X modified_with_respect_to A) by Th102;
    assume that
A3: W is open and
A4: f.x1 in W;
    W in the topology of (X modified_with_respect_to X0) by A3;
    then W in A-extension_of_the_topology_of X by Th102;
    then
A5: W0 is open;
    thus ex V being Subset of X1 st V is open & x1 in V & f.:V c= W
    proof
      consider V being Subset of X1 such that
A6:   V is open & x1 in V & ((modid(X,A))|X1).:V c= W0 by A2,A4,A5,Th43;
      take V;
      thus thesis by A6;
    end;
  end;
  then f is_continuous_at x1 by Th43;
  hence thesis by Def11;
end;
