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 Th28:
  #Z n `| Z = (n(#) #Z (n-1)) | Z
proof
  dom (n (#) #Z (n-1) ) = dom ( #Z (n-1) ) by VALUED_1:def 5;
  then
A1: dom (n (#) #Z (n-1) ) = REAL by FUNCT_2:def 1;
  then n (#) #Z (n-1) is Function of REAL, REAL by FUNCT_2:67;
  then
A2: dom ((n (#) #Z (n-1)) | Z) = Z by Th1;
A3: #Z n is_differentiable_on Z by Th8,FDIFF_1:26;
A4: for x be Element of REAL st x in Z
    holds (( #Z n ) `| Z ).x = ((n (#) #Z (n-1) )| Z
  ).x
  proof
    let x be Element of REAL such that
A5: x in Z;
    (( #Z n ) `| Z).x = diff(( #Z n),x) by A3,A5,FDIFF_1:def 7
      .= n * x #Z (n-1) by TAYLOR_1:2
      .=n*( #Z (n-1) ).x by TAYLOR_1:def 1
      .=(n (#) #Z (n-1)).x by A1,VALUED_1:def 5
      .=((n (#) #Z (n-1)) | Z).x by A2,A5,FUNCT_1:47;
    hence thesis;
  end;
  dom (( #Z n )`| Z) = Z by A3,FDIFF_1:def 7;
  hence thesis by A2,A4,PARTFUN1:5;
end;
