reserve i,n,m for Nat;

theorem Th10:
for m,n be non zero Nat
for 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
    Point of R_NormSpace_of_BoundedLinearOperators(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
   ex g be PartFunc of REAL-NS m,REAL-NS n, y be Point of REAL-NS m st
    f=g & x=y & diff(f,x) = diff(g,y) by PDIFF_1:def 8;
   hence thesis;
end;
