
theorem Th8:
  for n being Ordinal, L being add-associative right_complementable
right_zeroed well-unital distributive domRing-like non trivial doubleLoopStr,
  p being Polynomial of n,L, m being non-zero Monomial of n,L, b being bag of n
  holds b in Support(p) iff term(m) + b in Support(m*'p)
proof
  let n be Ordinal, L be add-associative right_complementable right_zeroed
  well-unital distributive domRing-like non trivial doubleLoopStr, p be
  Polynomial of n,L, m be non-zero Monomial of n,L, b be bag of n;
A1: (m*'p).(term(m) + b) = m.term(m) * p.b by POLYRED:7;
  m <> 0_(n,L) by POLYNOM7:def 1;
  then Support m <> {} by POLYNOM7:1;
  then Support m = {term(m)} by POLYNOM7:7;
  then term(m) in Support m by TARSKI:def 1;
  then
A2: m.term(m) <> 0.L by POLYNOM1:def 4;
A3: now
    assume b in Support p;
    then p.b <> 0.L by POLYNOM1:def 4;
    then
    term(m) + b is Element of Bags n & (m*'p).(term(m) + b) <> 0.L by A2,A1,
PRE_POLY:def 12,VECTSP_2:def 1;
    hence term(m) + b in Support(m*'p) by POLYNOM1:def 4;
  end;
  now
    assume term(m) + b in Support(m*'p);
    then m.term(m) * p.b <> 0.L by A1,POLYNOM1:def 4;
    then
A4: p.b <> 0.L;
    b is Element of Bags n by PRE_POLY:def 12;
    hence b in Support p by A4,POLYNOM1:def 4;
  end;
  hence thesis by A3;
end;
