reserve x,r,a,x0,p for Real;
reserve n,i,m for Element of NAT;
reserve Z for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve k for Nat;

theorem Th17:
  f1 is_differentiable_on n,Z & f2 is_differentiable_on n,Z & i<=n
  implies diff(f1+f2,Z).i = diff(f1,Z).i + diff(f2,Z).i
proof
  assume
A1: f1 is_differentiable_on n,Z & f2 is_differentiable_on n,Z;
  assume i<=n;
  then f1 is_differentiable_on i, Z & f2 is_differentiable_on i, Z by A1,
TAYLOR_1:23;
  hence thesis by Th15;
end;
