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 Th35:
  for f1,f2,x0 st f1 is_differentiable_in x0 & f2
  is_differentiable_in x0 holds f1+f2 is_differentiable_in x0 & diff(f1+f2,x0)=
  diff(f1,x0)+diff(f2,x0)
proof
  let f1,f2,x0;
  assume that
A1: f1 is_differentiable_in x0 and
A2: f2 is_differentiable_in x0;
  consider N1 be Neighbourhood of x0 such that
A3: N1 c= dom f1 and
A4: ex L,R st for x be Point of S st x in N1 holds f1/.x - f1/.x0 = L.(x
  -x0) + R/.(x-x0) by A1;
  consider L1,R1 such that
A5: for x be Point of S st x in N1 holds f1/.x - f1/.x0 = L1.(x-x0) + R1
  /.(x-x0) by A4;
  consider N2 be Neighbourhood of x0 such that
A6: N2 c= dom f2 and
A7: ex L,R st for x be Point of S st x in N2 holds f2/.x - f2/.x0 = L.(x
  -x0) + R/.(x-x0) by A2;
  consider L2,R2 such that
A8: for x be Point of S st x in N2 holds f2/.x - f2/.x0 = L2.(x-x0) + R2
  /.(x-x0) by A7;
  reconsider R=R1+R2 as RestFunc of S,T by Th28;
  set L=L1+L2;
  consider N be Neighbourhood of x0 such that
A9: N c= N1 and
A10: N c= N2 by Th1;
A11: N c= dom f2 by A6,A10;
  N c= dom f1 by A3,A9;
  then N /\ N c= dom f1 /\ dom f2 by A11,XBOOLE_1:27;
  then
A12: N c= dom (f1+f2) by VFUNCT_1:def 1;
A13: R1 is total & R2 is total by Def5;
A14: now
    let x be Point of S;
A15: x0 in N by NFCONT_1:4;
    assume
A16: x in N;
    hence (f1+f2)/.x - (f1+f2)/.x0 = (f1/.x+f2/.x) - (f1+f2)/.x0 by A12,
VFUNCT_1:def 1
      .=f1/.x+f2/.x - (f1/.x0+f2/.x0) by A12,A15,VFUNCT_1:def 1
      .=f1/.x+f2/.x - f1/.x0 - f2/.x0 by RLVECT_1:27
      .=(f1/.x +-f1/.x0) + f2/.x - f2/.x0 by RLVECT_1:def 3
      .=(f1/.x - f1/.x0) + (f2/.x - f2/.x0) by RLVECT_1:def 3
      .=L1.(x-x0)+R1/.(x-x0) + (f2/.x - f2/.x0) by A5,A9,A16
      .=L1.(x-x0)+R1/.(x-x0) + (L2.(x-x0) + R2/.(x-x0)) by A8,A10,A16
      .=R1/.(x-x0)+L1.(x-x0) + L2.(x-x0) + R2/.(x-x0) by RLVECT_1:def 3
      .=(L1.(x-x0)+L2.(x-x0)) + R1/.(x-x0) + R2/.(x-x0) by RLVECT_1:def 3
      .=(L1.(x-x0)+L2.(x-x0)) + (R1/.(x-x0) + R2/.(x-x0)) by RLVECT_1:def 3
      .=L.(x-x0)+(R1/.(x-x0) + R2/.(x-x0)) by LOPBAN_1:35
      .=L.(x-x0)+R/.(x-x0) by A13,VFUNCT_1:37;
  end;
  hence f1+f2 is_differentiable_in x0 by A12;
  hence diff(f1+f2,x0)=L by A12,A14,Def7
    .=diff(f1,x0) + L2 by A1,A3,A5,Def7
    .=diff(f1,x0) + diff(f2,x0) by A2,A6,A8,Def7;
end;
