theorem Th47:
  bdif(f,h).1.x = bdif(f,h).0.x-bdif(f,h).0.(x-h)
proof
  bdif(f,h).1.x = bD(f,h).x by DIFF_3:11
    .= f.x-f.(x-h) by DIFF_1:4
    .= bdif(f,h).0.x-f.(x-h) by DIFF_1:def 7
    .= bdif(f,h).0.x-bdif(f,h).0.(x-h) by DIFF_1:def 7;
  hence thesis;
end;
