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 Th8:
  #Z n is_differentiable_on REAL
proof
A1: dom( #Z n) = REAL by FUNCT_2:def 1;
  for x st x in REAL holds ( #Z n)|REAL is_differentiable_in x
  proof
    let x        ;
    assume x in REAL;
    #Z n is_differentiable_in x by TAYLOR_1:2;
    hence thesis by A1,RELAT_1:68;
  end;
  hence thesis by A1,FDIFF_1:def 6;
end;
