reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve y,X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1,p2 for Real;
reserve i,m,k for Element of NAT;
reserve n,k for non zero Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq,seq1 for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve g,g1,g2 for PartFunc of REAL,REAL-NS n;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;

theorem Th11:
  f1 is_differentiable_in x0 & f2 is_differentiable_in x0 implies
  f1+f2 is_differentiable_in x0 & diff(f1+f2,x0) = diff(f1,x0)+diff(f2,x0)
proof
  assume
A1: f1 is_differentiable_in x0 & f2 is_differentiable_in x0;
  consider g1 be PartFunc of REAL,REAL-NS n such that
A2: f1=g1 & g1 is_differentiable_in x0 by A1;
  consider g2 be PartFunc of REAL,REAL-NS n such that
A3: f2=g2 & g2 is_differentiable_in x0 by A1;
A4: g1+g2 is_differentiable_in x0
     & diff(g1+g2,x0)=diff(g1,x0)+diff(g2,x0) by A2,A3,NDIFF_3:14;
A5: f1+f2 = g1+g2 by A2,A3,NFCONT_4:5;
A6: diff(f1,x0) = diff(g1,x0) & diff(f2,x0) = diff(g2,x0) by A2,A3,Th3;
  diff(f1+f2,x0) = diff(g1+g2,x0) by A2,A3,Th3,NFCONT_4:5;
  hence thesis by A5,A4,A6,REAL_NS1:2;
end;
