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;

theorem Th81:
  the TopStruct of X1 = X0 implies for x0 being Point of X0, x1
  being Point of X1 st x0 = x1 holds g|X1 is_continuous_at x1 implies g
  is_continuous_at x0
proof
  reconsider Y1 = the TopStruct of X1 as TopSpace;
  assume
A1: the TopStruct of X1 = X0;
  then the carrier of X1 c= the carrier of X0;
  then reconsider A = the carrier of X1 as Subset of X0;
  A = [#]X0 by A1;
  then
A2: A is open;
  Y1 is SubSpace of X0 by A1,TSEP_1:2;
  then
A3: X1 is open SubSpace of X0 by A2,Th7,TSEP_1:16;
  let x0 be Point of X0, x1 be Point of X1 such that
A4: x0 = x1;
  assume g|X1 is_continuous_at x1;
  hence thesis by A4,A3,Th79;
end;
