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;

theorem Th31:
 (bdif(f1(#)f2,h).1).x = f1.x* (bdif(f2,h).1).x + f2.(x-h)* (bdif(f1,h).1).x
proof
  (bdif(f1(#)f2,h).1).x = (bdif(f1(#)f2,h).(0+1)).x
    .= bD(bdif(f1(#)f2,h).0,h).x by DIFF_1:def 7
    .= bD(f1(#)f2,h).x by DIFF_1:def 7
    .= (f1(#)f2).x - (f1(#)f2).(x-h) by DIFF_1:4
    .= f1.x* f2.x - (f1(#)f2).(x-h) by VALUED_1:5
    .= f1.x* f2.x - f1.(x-h)* f2.(x-h) by VALUED_1:5
    .= f1.x* (f2.x-f2.(x-h)) + f2.(x-h)* (f1.x-f1.(x-h))
    .= f1.x* bD(f2,h).x + f2.(x-h)* (f1.x-f1.(x-h)) by DIFF_1:4
    .= f1.x* bD(f2,h).x + f2.(x-h)* bD(f1,h).x by DIFF_1:4
    .= f1.x* bD(bdif(f2,h).0,h).x + f2.(x-h)* bD(f1,h).x by DIFF_1:def 7
    .= f1.x* bD(bdif(f2,h).0,h).x
       + f2.(x-h)* bD(bdif(f1,h).0,h).x by DIFF_1:def 7
    .= f1.x* bdif(f2,h).(0+1).x
       + f2.(x-h)* bD(bdif(f1,h).0,h).x by DIFF_1:def 7
    .= f1.x* (bdif(f2,h).1).x + f2.(x-h)* (bdif(f1,h).1).x by DIFF_1:def 7;
  hence thesis;
end;
