reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem
for n be Nat, f,g be PartFunc of S,T
 st 1 <= n & f is_differentiable_on n,Z & g is_differentiable_on n,Z
 holds f+g is_differentiable_on n,Z
proof
   let n be Nat, f,g be PartFunc of S,T;
   assume A1: 1<= n & f is_differentiable_on n,Z
           & g is_differentiable_on n,Z; then
A2: Z is open by Th18;
   Z /\ Z c= (dom f) /\ (dom g) by A1,XBOOLE_1:19; then
A3:Z c= dom (f+g) by VFUNCT_1:def 1;
   for i be Nat st i <= n-1 holds
    diff(f+g,i,Z) is_differentiable_on Z
   proof
    let i be Nat;
    assume A4: i <= n-1; then
A5: diff(f,i,Z) is_differentiable_on Z & diff(g,i,Z) is_differentiable_on Z
      by Th14,A1;
    n-1 <= n by XREAL_1:43; then
A6: i <= n by A4,XXREAL_0:2; then
    dom diff(f,i,Z) = Z & dom diff(g,i,Z) = Z by Th19,A1; then
    Z /\ Z c= dom diff(f,i,Z) /\ dom diff(g,i,Z); then
    Z c= dom (diff(f,i,Z) + diff(g,i,Z)) by VFUNCT_1:def 1; then
    diff(f,i,Z) + diff(g,i,Z) is_differentiable_on Z by A2,A5,NDIFF_1:39;
    hence diff(f+g,i,Z) is_differentiable_on Z by A1,A6,Th20;
   end;
   hence thesis by Th14,A3;
end;
