
theorem Th9:
  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 holds Support(m*'p)
  = { term(m) + b where b is Element of Bags n : b in Support 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;
  m <> 0_(n,L) by POLYNOM7:def 1;
  then Support m <> {} by POLYNOM7:1;
  then
A1: Support m = {term(m)} by POLYNOM7:7;
A2: Support(m*'p) c= {s + t where s,t is Element of Bags n : s in Support m
  & t in Support p} by TERMORD:30;
A3: now
    let u be object;
    assume
A4: u in Support(m*'p);
    then reconsider u9 = u as Element of Bags n;
    u9 in {s + t where s,t is Element of Bags n : s in Support m & t in
    Support p} by A2,A4;
    then consider s,t being Element of Bags n such that
A5: u9 = s + t & s in Support m and
A6: t in Support p;
    u9 = term(m) + t by A1,A5,TARSKI:def 1;
    hence u in {term(m)+b where b is Element of Bags n : b in Support p} by A6;
  end;
  now
    let u be object;
    assume u in {term(m)+b where b is Element of Bags n : b in Support p};
    then ex t being Element of Bags n st u = term(m) + t & t in Support p;
    hence u in Support(m*'p) by Th8;
  end;
  hence thesis by A3,TARSKI:2;
end;
