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 Th43:
  not 0 in Z implies ( #Z n)^ is_differentiable_on Z
proof
  assume
A1: not 0 in Z;
A2: for x0 st x0 in Z holds #Z n.x0 <> 0
  proof
    let x0;
A3: #Z n.x0 = x0 #Z n by TAYLOR_1:def 1;
    assume x0 in Z;
    hence thesis by A1,A3,PREPOWER:38;
  end;
  #Z n is_differentiable_on Z by Th8,FDIFF_1:26;
  hence thesis by A2,FDIFF_2:22;
end;
