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
  x in Z & n>m implies (diff(a(#) #Z n,Z).m).x = a*(n choose m)*(m!)*x #Z (n-m)
proof
  assume that
A1: x in Z and
A2: n>m;
  dom( #Z (n-m))=REAL by FUNCT_2:def 1;
  then
A3: dom((n choose m)*(m!)(#) #Z (n-m))= REAL by VALUED_1:def 5;
A4: dom(diff( #Z n,Z).m) = dom(((n choose m)*(m!)(#) #Z (n-m)) | Z) by A2,Th32
    .= REAL /\ Z by A3,RELAT_1:61
    .= Z by XBOOLE_1:28;
A5: dom(a(#)diff( #Z n,Z).m) = dom(diff( #Z n,Z).m) by VALUED_1:def 5;
  #Z n is_differentiable_on m, Z by A2,Th38,TAYLOR_1:23;
  then (diff(a(#) #Z n,Z).m).x = (a(#)diff( #Z n,Z).m).x by Th21
    .= a*(diff( #Z n,Z).m).x by A1,A4,A5,VALUED_1:def 5
    .= a*((n choose m)*(m!)*x #Z (n-m)) by A1,A2,Th36;
  hence thesis;
end;
