reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1 for sequence of S;
reserve x0 for Point of S;
reserve h for (0.S)-convergent sequence of S;
reserve c for constant sequence of S;
reserve R,R1,R2 for RestFunc of S,T;
reserve L,L1,L2 for Point of R_NormSpace_of_BoundedLinearOperators(S,T);

theorem
  for Z be Subset of S st Z is open for f1,f2 st Z c= dom (f1+f2) & f1
  is_differentiable_on Z & f2 is_differentiable_on Z holds f1+f2
is_differentiable_on Z & for x be Point of S st x in Z holds ((f1+f2)`|Z)/.x =
  diff(f1,x) + diff(f2,x)
proof
  let Z be Subset of S such that
A1: Z is open;
  let f1,f2;
  assume that
A2: Z c= dom (f1+f2) and
A3: f1 is_differentiable_on Z & f2 is_differentiable_on Z;
  now
    let x0 be Point of S;
    assume x0 in Z;
    then f1 is_differentiable_in x0 & f2 is_differentiable_in x0 by A1,A3,Th31;
    hence f1+f2 is_differentiable_in x0 by Th35;
  end;
  hence
A4: f1+f2 is_differentiable_on Z by A1,A2,Th31;
  now
    let x be Point of S;
    assume
A5: x in Z;
    then
A6: f1 is_differentiable_in x & f2 is_differentiable_in x by A1,A3,Th31;
    thus ((f1+f2)`|Z)/.x = diff((f1+f2),x) by A4,A5,Def9
      .= diff(f1,x) + diff(f2,x) by A6,Th35;
  end;
  hence thesis;
end;
