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 B being Subset of Y st g"B c= the
  carrier of X1 holds g"B = (g|X1)"B
proof
  assume
A1: X1 is SubSpace of X0;
  let B be Subset of Y;
  assume g"B c= the carrier of X1;
  hence g"B = (g|(the carrier of X1))"B by FUNCT_2:98
    .= (g|X1)"B by A1,Def5;
end;
