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 Th7:
  n>=1 & f1 is_differentiable_on n,Z implies f1 is_differentiable_on Z
proof
  assume that
A1: n>=1 and
A2: f1 is_differentiable_on n,Z;
  1-1<=n-1 by A1,XREAL_1:9;
  then diff(f1,Z).0 is_differentiable_on Z by A2;
  then f1|Z is_differentiable_on Z by TAYLOR_1:def 5;
  hence thesis by Th6;
end;
