reserve n,m for Element of NAT;
reserve h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
reserve f,f1,f2 for Function of REAL,REAL;
reserve S for Seq_Sequence;

theorem
  (for n, i be Nat st i<=n holds
  (S.n).i=(n choose i) * bdif(f1,h).i.x * bdif(f2,h).(n-'i).(x-i*h)) implies
  bdif(f1(#)f2,h).1.x = Sum(S.1, 1) & bdif(f1(#)f2,h).2.x = Sum(S.2, 2)
proof
  assume
A1: for n,i be Nat st i<=n holds
  S.n.i=(n choose i) * bdif(f1,h).i.x * bdif(f2,h).(n-'i).(x-i*h);
A2: 1-'0 = 1-0 by XREAL_1:233
      .= 1;
A3: (S.1).0 = (1 choose 0) * (bdif(f1,h).0).x * bdif(f2,h).(1-'0).(x-0*h) by A1
      .= 1 * (bdif(f1,h).0).x * bdif(f2,h).(1-'0).(x-0*h) by NEWTON:19
      .= f1.x * bdif(f2,h).1.x by A2,DIFF_1:def 7;
A4: 1-'1 = 1-1 by XREAL_1:233
      .= 0;
A5: (S.1).1 = (1 choose 1) * (bdif(f1,h).1).x * bdif(f2,h).(1-'1).(x-1*h) by A1
      .= 1 * (bdif(f1,h).1).x * bdif(f2,h).(1-'1).(x-1*h) by NEWTON:21
      .= f2.(x-h) * bdif(f1,h).1.x by A4,DIFF_1:def 7;
A6: 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
    .= (bdif(f1(#)f2,h).1).x by A3,A5,Th31;
A7: bdif(f1(#)f2,h).1 is Function of REAL,REAL by DIFF_1:12;
A8: bdif(f2,h).1 is Function of REAL,REAL by DIFF_1:12;
A9: bdif(f1,h).1 is Function of REAL,REAL by DIFF_1:12;
A10: bdif(f1(#)f2,h).2.x = bdif(f1(#)f2,h).(1+1).x
    .= bD(bdif(f1(#)f2,h).1,h).x by DIFF_1:def 7
    .= bdif(f1(#)f2,h).1.x - bdif(f1(#)f2,h).1.(x-h) by A7,DIFF_1:4
    .= f1.x * bdif(f2,h).1.x + bdif(f1,h).1.x * f2.(x-h)
       - bdif(f1(#)f2,h).1.(x-h) by Th31
    .= f1.x * bdif(f2,h).1.x + bdif(f1,h).1.x * f2.(x-h)
       - (f1.(x-h) * (bdif(f2,h).1).(x-h)
       + (bdif(f1,h).1).(x-h) * f2.((x-h)-h)) by Th31
    .= f1.x * (bdif(f2,h).1.x - bdif(f2,h).1.(x-h))
       + (f1.x - f1.(x-h)) * bdif(f2,h).1.(x-h)
       - (bdif(f1,h).1.(x-h) - bdif(f1,h).1.x) * f2.(x-2*h)
       - bdif(f1,h).1.x * (f2.(x-2*h) - f2.(x-h))
    .= f1.x * bD(bdif(f2,h).1,h).x + (f1.x - f1.(x-h)) * bdif(f2,h).1.(x-h)
       - (bdif(f1,h).1.(x-h) - bdif(f1,h).1.x) * f2.(x-2*h)
       - bdif(f1,h).1.x * (f2.(x-2*h) - f2.(x-h)) by A8,DIFF_1:4
    .= f1.x * bD(bdif(f2,h).1,h).x + bD(f1,h).x * bdif(f2,h).1.(x-h)
       + ((bdif(f1,h).1).x - (bdif(f1,h).1).(x-h)) * f2.(x-2*h)
       - bdif(f1,h).1.x * (f2.(x-2*h) - f2.(x-h)) by DIFF_1:4
    .= (f1.x * bD(bdif(f2,h).1,h).x + bD(f1,h).x * bdif(f2,h).1.(x-h))
       + bD(bdif(f1,h).1,h).x * f2.(x-2*h)
       + bdif(f1,h).1.x * (f2.(x-h) - f2.((x-h)-h)) by A9,DIFF_1:4
    .= (f1.x * bD(bdif(f2,h).1,h).x + bD(f1,h).x * bdif(f2,h).1.(x-h))
       + bD(bdif(f1,h).1,h).x * f2.(x-2*h)
       + bdif(f1,h).1.x * bD(f2,h).(x-h) by DIFF_1:4
    .= (f1.x * bdif(f2,h).(1+1).x
       + bD(f1,h).x * bdif(f2,h).1.(x-h))
       + bD(bdif(f1,h).1,h).x * f2.(x-2*h)
       + bdif(f1,h).1.x * bD(f2,h).(x-h) by DIFF_1:def 7
    .= (f1.x * bdif(f2,h).(1+1).x
       + bD(bdif(f1,h).0,h).x * bdif(f2,h).1.(x-h))
       + bD(bdif(f1,h).1,h).x * f2.(x-2*h)
       + bdif(f1,h).1.x * bD(f2,h).(x-h) by DIFF_1:def 7
    .= (f1.x * bdif(f2,h).2.x
       + bD(bdif(f1,h).0,h).x * bdif(f2,h).1.(x-h))
       + bdif(f1,h).2.x * f2.(x-2*h)
       + bdif(f1,h).1.x * bD(f2,h).(x-h) by DIFF_1:def 7
    .= (f1.x * bdif(f2,h).2.x
       + bdif(f1,h).(0+1).x * bdif(f2,h).1.(x-h))
       + bdif(f1,h).2.x * f2.(x-2*h)
       + bdif(f1,h).1.x * bD(f2,h).(x-h) by DIFF_1:def 7
    .= (f1.x * bdif(f2,h).2.x
       + bdif(f1,h).1.x * bdif(f2,h).1.(x-h))
       + bdif(f1,h).2.x * f2.(x-2*h)
       + bdif(f1,h).1.x * bD(bdif(f2,h).0,h).(x-h) by DIFF_1:def 7
    .= (f1.x * bdif(f2,h).2.x
       + bdif(f1,h).1.x * bdif(f2,h).1.(x-h))
       + bdif(f1,h).2.x * f2.(x-2*h)
       + bdif(f1,h).1.x * bdif(f2,h).(0+1).(x-h) by DIFF_1:def 7
    .= f1.x * bdif(f2,h).2.x
       + 2 * (bdif(f1,h).1.x * bdif(f2,h).1.(x-h))
       + bdif(f1,h).2.x * f2.(x-2*h);
A11: 2-'0 = 2-0 by XREAL_1:233
       .= 2;
A12: (S.2).0 = (2 choose 0) * (bdif(f1,h).0).x
               * bdif(f2,h).(2-'0).(x-0*h) by A1
       .= 1 * (bdif(f1,h).0).x * bdif(f2,h).(2-'0).(x-0*h) by NEWTON:19
       .= f1.x * bdif(f2,h).2.x by A11,DIFF_1:def 7;
A13: 2-'1 = 2-1 by XREAL_1:233
       .= 1;
A14: (S.2).1 = (2 choose 1) * (bdif(f1,h).1).x
               * bdif(f2,h).(2-'1).(x-1*h) by A1
       .= 2 * (bdif(f1,h).1).x * bdif(f2,h).1.(x-h) by A13,NEWTON:23;
A15: 2-'2 = 2-2 by XREAL_1:233
       .= 0;
A16: (S.2).2 = (2 choose 2) * (bdif(f1,h).2).x
               * bdif(f2,h).(2-'2).(x-2*h) by A1
       .= 1 * (bdif(f1,h).2).x * bdif(f2,h).(2-'2).(x-2*h) by NEWTON:21
       .= bdif(f1,h).2.x * f2.(x-2*h) by A15,DIFF_1:def 7;
  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
    .= bdif(f1(#)f2,h).2.x by A10,A12,A14,A16,SERIES_1:def 1;
  hence thesis by A6;
end;
