
theorem Th15:
  for n being Ordinal, T being connected TermOrder of n, L being
add-associative right_complementable right_zeroed non trivial doubleLoopStr,
  p being Polynomial of n,L holds p - Red(p,T) = HM(p,T)
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
right_complementable right_zeroed non trivial doubleLoopStr, p be Polynomial
  of n,L;
  thus p - Red(p,T) = (HM(p,T) + Red(p,T)) - Red(p,T) by TERMORD:38
    .= (HM(p,T) + Red(p,T)) + (-Red(p,T)) by POLYNOM1:def 7
    .= HM(p,T) + (Red(p,T) + (-Red(p,T))) by POLYNOM1:21
    .= HM(p,T) + 0_(n,L) by POLYRED:3
    .= HM(p,T) by POLYNOM1:23;
end;
