theorem Th48:
  poly_diff(-p) = -poly_diff(p)
  proof
    let n be Element of NAT;
A1: (poly_diff(p)).n = p.(n+1) * (n+1) by Def5;
    dom(-p) = NAT by FUNCT_2:def 1;
    then
A2: (-p)/.(n+1) = -(p/.(n+1)) by VFUNCT_1:def 5;
A3: dom(-poly_diff(p)) = NAT by FUNCT_2:def 1;
    thus (poly_diff(-p)).n = (-p).(n+1) * (n+1) by Def5
    .= -((poly_diff(p))/.n) by A1,A2
    .= (-poly_diff(p))/.n by A3,VFUNCT_1:def 5
    .= (-poly_diff(p)).n;
  end;
