reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
          right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
             right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;

theorem Th43:
  for X be Ordinal, S be add-associative right_complementable right_zeroed
    right_unital distributive non empty doubleLoopStr
  for p,q be Polynomial of X, S holds
    vars (p*'q) c= vars p \/  vars q
proof
  let X be Ordinal, S be add-associative right_complementable right_zeroed
    right_unital distributive non empty doubleLoopStr;
  let p,q be Polynomial of X, S;
  let x;
  assume x in vars (p*'q);
  then consider b be bag of X such that
A1: b in Support (p*'q) & b.x <> 0 by Def5;
  reconsider b as Element of Bags X by PRE_POLY:def 12;
  consider s be FinSequence of the carrier of S such that
A2:(p*'q).b = Sum s and
A3:len s = len decomp b and
A4:for k be Element of NAT st k in dom s ex b1, b2 be bag of
  X st (decomp b)/.k = <*b1, b2*> & s/.k = (p.b1) * (q.b2)
  by POLYNOM1:def 10;
  (p*'q).b <> 0.S by A1,POLYNOM1:def 4;
  then consider k be Nat such that
A5:k in dom s and
A6:s/.k <> 0.S by A2,MATRLIN:11;
  consider b1, b2 be bag of X such that
A7:(decomp b)/.k = <*b1, b2*> and
A8:s/.k = p.b1*q.b2 by A4,A5;
A9:b1 in Bags X by PRE_POLY:def 12;
A10:b2 in Bags X by PRE_POLY:def 12;
  q.b2 <> 0.S by A6,A8;
  then
A11:b2 in Support q by A10,POLYNOM1:def 4;
  p.b1 <> 0.S by A6,A8;
  then
A12:b1 in Support p by A9,POLYNOM1:def 4;
  k in dom decomp b by A3,A5,FINSEQ_3:29;
  then consider b19, b29 be bag of X such that
A13:(decomp b)/.k = <*b19, b29*> and
A14:b = b19+b29 by PRE_POLY:68;
  b19 = b1 & b29 = b2 by A7,A13,FINSEQ_1:77;
  then b.x = b1.x+b2.x by A14,PRE_POLY:def 5;
  then b1.x <>0 or b2.x <>0 by A1;
  then x in vars p or x in vars q by A11,A12,Def5;
  hence thesis by XBOOLE_0:def 3;
end;
