
theorem Th24:
  for m be non zero Nat
  for f be PartFunc of REAL m,REAL
  for x be Element of REAL m
    st f is_differentiable_in x
  holds
    for u be Element of REAL m, a be Real
    holds diff(f,x).(a*u) = a * diff(f,x).u
proof
  let m be non zero Nat;
  let f be PartFunc of (REAL m),REAL;
  let x be Element of REAL m;

  assume
  A1: f is_differentiable_in x;

  let u be Element of REAL m;
  let a be Real;
  A2: <>* f is_differentiable_in x by A1,PDIFF_7:def 1;

  consider g be PartFunc of REAL-NS m,REAL-NS 1,
  y be Point of REAL-NS m such that
  A3: <>* f = g & x = y & diff(<>* f, x) = diff(g,y)
      by A2,PDIFF_1:def 8;

  reconsider DF = diff(g,y) as
    Lipschitzian LinearOperator of REAL-NS m,REAL-NS 1
    by LOPBAN_1:def 9;

  A4: dom(diff(<>* f,x))
    = dom DF by A3
  .= the carrier of REAL-NS m by FUNCT_2:def 1;

  A5: the carrier of REAL-NS m = REAL m by REAL_NS1:def 4;
  reconsider u1 = u as Point of REAL-NS m by REAL_NS1:def 4;

  thus diff(f,x).(a * u)
    = (proj(1,1) * diff(<>* f, x)).(a * u) by PDIFF_7:def 2
  .= proj(1,1).(diff(<>* f,x).(a * u)) by A4,A5,FUNCT_1:13
  .= proj(1,1).(DF.(a * u1)) by REAL_NS1:3,A3
  .= proj(1,1).(a * DF.u1) by LOPBAN_1:def 5
  .= a * proj(1,1).(DF.u1) by PDIFF_1:4
  .= a * (proj(1,1) * diff(<>* f,x)).u by A3,A4,FUNCT_1:13
  .= a * diff(f,x).u by PDIFF_7:def 2;
end;
