
theorem mm5:
for R being non degenerated comRing
for p being Element of the carrier of Polynom-Ring R
for S being comRingExtension of R
for q being Element of the carrier of Polynom-Ring S
st q = p holds (Deriv S).q = (Deriv R).p
proof
let F be non degenerated comRing,
    p be Element of the carrier of Polynom-Ring F;
let E be comRingExtension of F,
    q be Element of the carrier of Polynom-Ring E;
assume AS: q = p;
the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E
  by FIELD_4:10; then
reconsider p1 = (Deriv F).p
                  as Element of the carrier of Polynom-Ring E;
now let o be object;
  assume o in NAT; then
  reconsider i = o as Element of NAT;
  ((Deriv E).q).i = (i+1) * q.(i+1) by RINGDER1:def 8
                 .= (i+1) * p.(i+1) by AS,mm5a
                 .= p1.i by RINGDER1:def 8;
  hence ((Deriv E).q).o = p1.o;
  end;
hence (Deriv E).q = (Deriv F).p by FUNCT_2:12;
end;
