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;

theorem
  X1 is SubSpace of X0 implies for g1 being Function of X1,Y st for x0
being Point of X0 st x0 in the carrier of X1 holds g.x0 = g1.x0 holds g|X1 = g1
proof
  assume
A1: X1 is SubSpace of X0;
  then
A2: the carrier of X1 is Subset of X0 by TSEP_1:1;
  let g1 be Function of X1,Y;
  assume
  for x0 being Point of X0 st x0 in the carrier of X1 holds g.x0 = g1. x0;
  then g|(the carrier of X1) = g1 by A2,FUNCT_2:96;
  hence thesis by A1,Def5;
end;
