reserve n,m,k,i for Nat,
  h,r,r1,r2,x0,x1,x2,x for Real,
  S for Functional_Sequence of REAL,REAL,
  y for set;
reserve f,f1,f2 for Function of REAL,REAL;
reserve S for Seq_Sequence;

theorem
  (for n for i st i<=n holds (S.n).i=(n choose i) * fdif(f1,h).i.x *
fdif(f2,h).(n-'i).(x+i*h)) implies fdif(f1(#)f2,h).1.x = Sum(S.1, 1) & fdif(f1
  (#)f2,h).2.x = Sum(S.2, 2)
proof
A1: 1-'0 = 1-0 by XREAL_1:233
    .= 1;
A2: 1-'1 = 1-1 by XREAL_1:233
    .= 0;
A3: fdif(f1(#)f2,h).1 is Function of REAL,REAL by Th2;
A4: 2-'1 = 2-1 by XREAL_1:233
    .= 1;
A5: fdif(f2,h).1 is Function of REAL,REAL by Th2;
A6: 2-'2 = 2-2 by XREAL_1:233
    .= 0;
  assume
A7: for n for i st i<=n holds S.n.i=(n choose i) * fdif(f1,h).i.x * fdif
  (f2,h).(n-'i).(x+i*h);
  then
A8: (S.2).1 = (2 choose 1) * (fdif(f1,h).1).x * fdif(f2,h).(2-'1).(x+1*h)
    .= 2 * (fdif(f1,h).1).x * fdif(f2,h).1.(x+h) by A4,NEWTON:23;
A9: (S.1).1 = (1 choose 1) * (fdif(f1,h).1).x * fdif(f2,h).(1-'1).(x+1*h) by A7
    .= 1 * (fdif(f1,h).1).x * fdif(f2,h).(1-'1).(x+1*h) by NEWTON:21
    .= fdif(f1,h).1.x * f2.(x+h) by A2,Def6;
A10: (S.1).0 = (1 choose 0) * (fdif(f1,h).0).x * fdif(f2,h).(1-'0).(x+0*h) by
A7
    .= 1 * (fdif(f1,h).0).x * fdif(f2,h).(1-'0).(x+0*h) by NEWTON:19
    .= f1.x * fdif(f2,h).1.x by A1,Def6;
A11: Sum(S.1, 1) = Partial_Sums(S.1).(0+1) by SERIES_1:def 5
    .= Partial_Sums(S.1).0 + S.1.1 by SERIES_1:def 1
    .= (S.1).0 + (S.1).1 by SERIES_1:def 1
    .= (fdif(f1(#)f2,h).1).x by A10,A9,Lm2;
A12: fdif(f1,h).1 is Function of REAL,REAL by Th2;
A13: fdif(f1(#)f2,h).2.x = fdif(f1(#)f2,h).(1+1).x
    .= fD(fdif(f1(#)f2,h).1,h).x by Def6
    .= fdif(f1(#)f2,h).1.(x+h) - fdif(f1(#)f2,h).1.x by A3,Th3
    .= f1.(x+h) * fdif(f2,h).1.(x+h) + fdif(f1,h).1.(x+h) * f2.(x+h+h) -
  fdif(f1(#)f2,h).1.x by Lm2
    .= f1.(x+h) * fdif(f2,h).1.(x+h) + fdif(f1,h).1.(x+h) * f2.(x+h+h) - (f1
  .x * fdif(f2,h).1.x + fdif(f1,h).1.x * f2.(x+h)) by Lm2
    .= f1.x * (fdif(f2,h).1.(x+h) - fdif(f2,h).1.x) + (f1.(x+h) - f1.x) *
fdif(f2,h).1.(x+h) + (fdif(f1,h).1.(x+h) - fdif(f1,h).1.x) * f2.(x+h+h) + fdif(
  f1,h).1.x * (f2.(x+h+h) - f2.(x+h))
    .= f1.x * fD(fdif(f2,h).1,h).x + (f1.(x+h) - f1.x) * fdif(f2,h).1.(x+h)
+ (fdif(f1,h).1.(x+h) - fdif(f1,h).1.x) * f2.(x+h+h) + fdif(f1,h).1.x * (f2.(x+
  h+h) - f2.(x+h)) by A5,Th3
    .= f1.x * fD(fdif(f2,h).1,h).x + fD(f1,h).x * fdif(f2,h).1.(x+h) + (fdif
(f1,h).1.(x+h) - fdif(f1,h).1.x) * f2.(x+h+h) + fdif(f1,h).1.x * (f2.(x+h+h) -
  f2.(x+h)) by Th3
    .= (f1.x * fD(fdif(f2,h).1,h).x + fD(f1,h).x * fdif(f2,h).1.(x+h)) + fD(
  fdif(f1,h).1,h).x * f2.(x+h+h) + fdif(f1,h).1.x * (f2.(x+h+h) - f2.(x+h)) by
A12,Th3
    .= (f1.x * fD(fdif(f2,h).1,h).x + fD(f1,h).x * fdif(f2,h).1.(x+h)) + fD(
  fdif(f1,h).1,h).x * f2.(x+h+h) + fdif(f1,h).1.x * fD(f2,h).(x+h) by Th3
    .= (f1.x * fdif(f2,h).(1+1).x + fD(f1,h).x * fdif(f2,h).1.(x+h)) + fD(
  fdif(f1,h).1,h).x * f2.(x+h+h) + fdif(f1,h).1.x * fD(f2,h).(x+h) by Def6
    .= (f1.x * fdif(f2,h).(1+1).x + fD(fdif(f1,h).0,h).x * fdif(f2,h).1.(x+h
  )) + fD(fdif(f1,h).1,h).x * f2.(x+h+h) + fdif(f1,h).1.x * fD(f2,h).(x+h) by
Def6
    .= (f1.x * fdif(f2,h).2.x + fD(fdif(f1,h).0,h).x * fdif(f2,h).1.(x+h)) +
  fdif(f1,h).2.x * f2.(x+2*h) + fdif(f1,h).1.x * fD(f2,h).(x+h) by Def6
    .= (f1.x * fdif(f2,h).2.x + fdif(f1,h).(0+1).x * fdif(f2,h).1.(x+h)) +
  fdif(f1,h).2.x * f2.(x+2*h) + fdif(f1,h).1.x * fD(f2,h).(x+h) by Def6
    .= (f1.x * fdif(f2,h).2.x + fdif(f1,h).1.x * fdif(f2,h).1.(x+h)) + fdif(
  f1,h).2.x * f2.(x+2*h) + fdif(f1,h).1.x * fD(fdif(f2,h).0,h).(x+h) by Def6
    .= (f1.x * fdif(f2,h).2.x + fdif(f1,h).1.x * fdif(f2,h).1.(x+h)) + fdif(
  f1,h).2.x * f2.(x+2*h) + fdif(f1,h).1.x * fdif(f2,h).(0+1).(x+h) by Def6
    .= f1.x * fdif(f2,h).2.x + 2 * (fdif(f1,h).1.x * fdif(f2,h).1.(x+h)) +
  fdif(f1,h).2.x * f2.(x+2*h);
A14: 2-'0 = 2-0 by XREAL_1:233
    .= 2;
A15: (S.2).2 = (2 choose 2) * (fdif(f1,h).2).x * fdif(f2,h).(2-'2).(x+2*h)
  by A7
    .= 1 * (fdif(f1,h).2).x * fdif(f2,h).(2-'2).(x+2*h) by NEWTON:21
    .= fdif(f1,h).2.x * f2.(x+2*h) by A6,Def6;
A16: (S.2).0 = (2 choose 0) * (fdif(f1,h).0).x * fdif(f2,h).(2-'0).(x+0*h)
  by A7
    .= 1 * (fdif(f1,h).0).x * fdif(f2,h).(2-'0).(x+0*h) by NEWTON:19
    .= f1.x * fdif(f2,h).2.x by A14,Def6;
  Sum(S.2, 2) = Partial_Sums(S.2).(1+1) by SERIES_1:def 5
    .= Partial_Sums(S.2).(0+1) + S.2.2 by SERIES_1:def 1
    .= Partial_Sums(S.2).0 + S.2.1 + S.2.2 by SERIES_1:def 1
    .= fdif(f1(#)f2,h).2.x by A13,A16,A8,A15,SERIES_1:def 1;
  hence thesis by A11;
end;
