
theorem Th10:
  for n being Ordinal, L being add-associative
  right_complementable left_zeroed 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 card Support(p) = card Support(m*'p)
proof
  let n be Ordinal, L be add-associative right_complementable left_zeroed
right_zeroed well-unital distributive domRing-like non trivial doubleLoopStr,
  p be Polynomial of n,L, m be non-zero Monomial of n,L;
  defpred P[object,object] means $2 = term(m) + In($1,Bags n);
  set T = the admissible connected TermOrder of n;
  m <> 0_(n,L) by POLYNOM7:def 1;
  then Support m <> {} by POLYNOM7:1;
  then
A1: Support m = {term(m)} by POLYNOM7:7;
A2: for x being object st x in Support(p)
  ex y being object st y in Support(m*'p) & P[x,y]
  proof
    let x be object;
    assume
A3: x in Support p;
    then reconsider x9 = x as Element of Bags n;
    term(m) + x9 in Support(m*'p) by A3,Th8;
    hence thesis;
  end;
  consider f being Function of Support(p),Support(m*'p) such that
A5: for x being object st x in Support(p) holds P[x,f.x] from FUNCT_2:sch 1
  (A2 );
A6: now
    assume
A7: Support(m*'p) = {};
    now
      assume Support p <> {};
      then p <> 0_(n,L) by POLYNOM7:1;
      then reconsider p9 = p as non-zero Polynomial of n,L by POLYNOM7:def 1;
      HT(m,T) + HT(p9,T) in Support(m*'p9) by TERMORD:29;
      hence contradiction by A7;
    end;
    hence Support(p) = {};
  end;
  then
A8: Support(p) c= dom f by FUNCT_2:def 1;
A9: 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;
A10: now
    let u be object;
    assume
A11: 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 A9,A11;
    then consider s,t being Element of Bags n such that
A12: u9 = s + t & s in Support m and
A13: t in Support p;
A14: t in dom f by A6,A13,FUNCT_2:def 1;
A15: t = In(t,Bags n);
    u9 = term(m) + t by A1,A12,TARSKI:def 1;
    then u9 = f.t by A5,A13,A15;
    hence u in f.:Support(p) by A14,FUNCT_1:def 6;
  end;
  now
    let x1,x2 be object;
    assume that
A16: x1 in Support(p) and
A17: x2 in Support(p) and
A18: f.x1 = f.x2;
    reconsider x19 = x1, x29 = x2 as Element of Bags n by A16,A17;
A19: x29 = In(x29,Bags n);
    x19 = In(x19,Bags n);
    then term(m) + x19 = f.x29 by A5,A16,A18
      .= term(m) + x29 by A5,A17,A19;
    hence x1 = x29 + term(m) -' term(m) by PRE_POLY:48
      .= x2 by PRE_POLY:48;
  end;
  then f is one-to-one by A6,FUNCT_2:19;
  then
A20: Support(p),f.:Support(p) are_equipotent by A8,CARD_1:33;
  for u being object holds u in f.:Support(p) implies u in Support(m*'p);
  then f.:Support(p) = Support(m*'p) by A10,TARSKI:2;
  hence thesis by A20,CARD_1:5;
end;
