reserve a,b,r for Real;
reserve A for non empty set;
reserve X,x for set;
reserve f,g,F,G for PartFunc of REAL,REAL;
reserve n for Element of NAT;

theorem Th29:
  #Z (n+1) is_integral_of (n+1)(#)( #Z n),REAL
proof
  reconsider m = n+1 as Nat;
A1: dom(((n+1)(#)( #Z n))|REAL) = REAL /\ REAL by FUNCT_2:def 1;
A2: dom( #Z (n+1)) = REAL by FUNCT_2:def 1;
  for x be Real
   st x in REAL holds ( #Z (n+1))|REAL is_differentiable_in x by TAYLOR_1:2;
  then
A3: #Z (n+1) is_differentiable_on REAL by A2;
A4: dom( #Z n) = REAL by FUNCT_2:def 1;
A5: now
    let x be object;
    assume
A6: x in dom(( #Z (n+1))`|REAL);
    then reconsider z=x as Real;
    (( #Z (n+1))`|REAL).x = diff(( #Z (n+1)),z) by A3,A6,FDIFF_1:def 7;
    then (( #Z (n+1))`|REAL).x = m * z #Z (m-1) by TAYLOR_1:2;
    then
A7: (( #Z (n+1))`|REAL).x = (n+1) * ( #Z n).x by TAYLOR_1:def 1;
    x in dom( #Z n) by A4,A6;
    then x in dom((n+1)(#)( #Z n)) by VALUED_1:def 5;
    then (( #Z (n+1))`|REAL).x = ((n+1)(#)( #Z n)).x by A7,VALUED_1:def 5;
    hence (( #Z (n+1))`|REAL).x = (((n+1)(#)( #Z n))|REAL).x;
  end;
  dom(( #Z (n+1))`|REAL) = REAL by A3,FDIFF_1:def 7;
  then (( #Z (n+1))`|REAL) = (((n+1)(#)( #Z n))|REAL) by A1,A5,FUNCT_1:2;
  hence thesis by A3,Lm1;
end;
