reserve a,b,r for Real;
reserve A for non empty set;
reserve X,x for set;
reserve f,g,F,G for PartFunc of REAL,REAL;
reserve n for Element of NAT;

theorem Th6:
  f is_differentiable_on X & g is_differentiable_on X implies
    f+g is_differentiable_on X & f-g is_differentiable_on X &
    f(#)g is_differentiable_on X
proof
  assume that
A1: f is_differentiable_on X and
A2: g is_differentiable_on X;
  reconsider Z = X as Subset of REAL by A1,FDIFF_1:8;
  reconsider Z as open Subset of REAL by A1,FDIFF_1:10;
  X c= dom f & X c= dom g by A1,A2;
  then
A3: Z c= dom f /\ dom g by XBOOLE_1:19;
  then
A4: Z c= dom(f(#)g) by VALUED_1:def 4;
  Z c= dom(f+g) & Z c= dom(f-g) by A3,VALUED_1:12,def 1;
  hence thesis by A1,A2,A4,FDIFF_1:18,19,21;
end;
