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 Th107:
  for X0 being non empty SubSpace of X for x0 being Point of X0
  holds (modid(X,X0))|X0 is_continuous_at x0
proof
  let X0 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))|X0 as Function of X0,X modified_with_respect_to
  X0 by Def10;
  let x0 be Point of X0;
A1: (modid(X,A))|X0 is_continuous_at x0 by Th98;
  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
A2: W is open and
A3: f.x0 in W;
    W in the topology of (X modified_with_respect_to X0) by A2;
    then W in A-extension_of_the_topology_of X by Th102;
    then
A4: W0 is open;
    thus ex V being Subset of X0 st V is open & x0 in V & f.:V c= W
    proof
      consider V being Subset of X0 such that
A5:   V is open & x0 in V & ((modid(X,A))|X0).:V c= W0 by A1,A3,A4,Th43;
      take V;
      thus thesis by A5;
    end;
  end;
  then f is_continuous_at x0 by Th43;
  hence thesis by Def11;
end;
