reserve i,n,m for Nat;

theorem Th18:
for m,n be non zero Nat,
    f be PartFunc of REAL m,REAL n, x be Element of REAL m st
  f is_differentiable_in x holds
    diff(f,x) is LinearOperator of REAL-NS m,REAL-NS n
proof
   let m,n be non zero Nat;
   let f be PartFunc of REAL m,REAL n,
       x be Element of REAL m;
   assume f is_differentiable_in x; then
   diff(f,x) is Point of R_NormSpace_of_BoundedLinearOperators
                     (REAL-NS m,REAL-NS n) by Th10;
   hence thesis by LOPBAN_1:def 9;
end;
