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 Th46:
  for X be Ordinal, S be right_zeroed add-associative right_complementable
    right_unital distributive well-unital non trivial doubleLoopStr
  for p be Polynomial of X, S, t be bag of X holds
    vars Subst(t,x,p) c= (support t)\{x} \/ vars p
proof
  let X be Ordinal, S be right_zeroed add-associative right_complementable
      right_unital distributive well-unital non trivial doubleLoopStr;
  let p be Polynomial of X, S, t be bag of X;
  let y be object;
  assume y in vars Subst(t,x,p);
  then consider b be bag of X such that
A1: b in Support Subst(t,x,p) & b.y <> 0 by Def5;
A2: Subst(t,x,p).b <>0.S by A1,POLYNOM1:def 3;
  then consider s be bag of X such that
A3: b = Subst(t,x,s) by Def3;
A4: y in dom b & dom b = X =dom t by A1,FUNCT_1:def 2,PARTFUN1:def 2;
A5: s in Bags X = dom (p `^ (t.x) ) by PRE_POLY:def 12,PARTFUN1:def 2;
  b.y = (t+*(x,0)).y + s.y by A3,PRE_POLY:def 5;
  then per cases by A1;
  suppose
A6: (t+*(x,0)).y<>0;
    then
A7: y<>x by A4,FUNCT_7:31;
    then (t+*(x,0)).y = t.y by FUNCT_7:32;
    then y in support t by A6,PRE_POLY:def 7;
    then y in (support t)\{x} by A7,ZFMISC_1:56;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
A8: s.y<>0;
    (p `^ (t.x) ).s <> 0.S by A2,A3,Def3;
    then s in Support (p `^ (t.x) ) by A5,POLYNOM1:def 4;
    then y in vars (p `^ (t.x) ) c= vars p by A8,Def5,Th45;
    hence thesis by XBOOLE_0:def 3;
  end;
end;
