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 Th36:
  n>m & x in Z implies (diff( #Z n,Z).m).x = (n choose m)*(m!)*x #Z (n-m)
proof
  assume that
A1: n>m and
A2: x in Z;
  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;
  then
A4: dom(((n choose m)*(m!)(#) #Z (n-m))|Z) = REAL /\ Z by RELAT_1:61
    .= Z by XBOOLE_1:28;
  reconsider xx=x as Element of REAL by XREAL_0:def 1;
  (diff( #Z n,Z).m).x =(((n choose m)*(m!)(#) #Z (n-m))| Z).x by A1,Th32
    .= ((n choose m)*(m!)(#) #Z (n-m)).x by A2,A4,FUNCT_1:47
    .=(n choose m)*(m!)*( #Z (n-m)).xx by A3,VALUED_1:def 5
    .=(n choose m)*(m!)*x #Z (n-m) by TAYLOR_1:def 1;
  hence thesis;
end;
