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
  for Y0 being SubSpace of X modified_with_respect_to X0 st the carrier
  of Y0 = the carrier of X0 holds Y0 is open SubSpace of X
  modified_with_respect_to X0
proof
  let Y0 be SubSpace of X modified_with_respect_to X0;
  assume
A1: the carrier of Y0 = the carrier of X0;
  reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
  set Y = X modified_with_respect_to X0;
  reconsider B = the carrier of Y0 as Subset of Y by TSEP_1:1;
  Y = X modified_with_respect_to A by Def10;
  then B is open by A1,Th94;
  hence thesis by TSEP_1:16;
end;
