
theorem Th23:
  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,v be Element of REAL m
    holds diff(f,x).(u+v) = diff(f,x).u + diff(f,x).v
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,v be Element of REAL m;
  A3: diff(f,x) = proj(1,1) * diff(<>* f, x) by PDIFF_7:def 2;
  <>* f is_differentiable_in x by A1,PDIFF_7:def 1; then

  consider g be PartFunc of REAL-NS m,REAL-NS 1,
            y be Point of REAL-NS m such that
  A4: <>* f = g & x = y & diff(<>* f, x) = diff(g,y) by PDIFF_1:def 8;

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

  A5: DF is additive;

  A6: dom(diff(<>* f,x))
    = dom DF by A4
  .= the carrier of REAL-NS m by FUNCT_2:def 1;

  A7: the carrier of REAL-NS m = REAL m by REAL_NS1:def 4;

  reconsider u1 = u,v1 = v as Point of REAL-NS m by REAL_NS1:def 4;

  thus diff(f,x).(u+v)
    =(proj(1,1) * diff(<>* f, x)).(u+v) by PDIFF_7:def 2
  .= proj(1,1).(diff(<>* f, x).(u+v)) by A6,A7,FUNCT_1:13
  .= proj(1,1).(DF.(u1 + v1)) by A4,REAL_NS1:2
  .= proj(1,1).(DF.u1 + DF.v1) by A5
  .= proj(1,1).(DF.u1) + proj(1,1).(DF.v1) by PDIFF_1:4
  .= (proj(1,1) * diff(<>* f, x)).u + proj(1,1).(diff(<>* f, x).v)
      by A4,A6,FUNCT_1:13
  .= diff(f,x).u + diff(f,x).v by A3,A6,A7,FUNCT_1:13;
end;
