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 Th83:
  X1 is SubSpace of X0 & X2 is SubSpace of X1 implies for g being
  Function of X0,Y holds g|X1 is continuous Function of X1,Y implies g|X2 is
  continuous Function of X2,Y
proof
  assume
A1: X1 is SubSpace of X0;
  assume
A2: X2 is SubSpace of X1;
  let g be Function of X0,Y;
  assume g|X1 is continuous Function of X1,Y;
  then (g|X1)|X2 is continuous Function of X2,Y by A2,Th82;
  hence thesis by A1,A2,Th72;
end;
