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 Th48:
  for X be set, S be non empty ZeroStr
    for s be Element of S holds vars Monom(s,(EmptyBag X) +*(x,n)) c= {x}
proof
  let X be set, S be non empty ZeroStr;
  let s be Element of S;
  set EX=(EmptyBag X) +*(x,n);
  let y be object;
  assume y in vars Monom(s,EX);
  then consider b be bag of X such that
A1: b in Support Monom(s,EX) & b.y <> 0 by Def5;
  assume not y in {x};
  then
A2: y <>x by TARSKI:def 1;
  Support Monom(s,EX) = {term(Monom(s,EX))} by A1,POLYNOM7:7;
  then
A3: b = term(Monom(s,EX)) by TARSKI:def 1,A1;
  coefficient Monom(s,EX)<>0.S by A1,POLYNOM7:def 5;
  then s<>0.S by POLYNOM7:8;
  then s is non zero & S is non trivial ZeroStr by STRUCT_0:def 12,def 18;
  then b=EX by A3,POLYNOM7:10;
  then b.y = (EmptyBag X).y by A2,FUNCT_7:32;
  hence thesis by A1;
end;
