reserve A for non degenerated comRing;
reserve R for non degenerated domRing;
reserve n for non empty Ordinal;
reserve o,o1,o2 for object;
reserve X,Y for Subset of Funcs(n,[#]R);
reserve S,T for Subset of Polynom-Ring(n,R);
reserve F,G for FinSequence of the carrier of Polynom-Ring(n,R);
reserve x for Function of n,R;

theorem Th11:
    for x be Element of n holds degree(UnitBag x) = 1
    proof
      let x be Element of n;
      set b = UnitBag x;
      dom b = n by PARTFUN1:def 2; then
A1:   x in dom b by SUBSET_1:def 1;
      dom (EmptyBag n) = n by PARTFUN1:def 2; then
A2:   x in dom(EmptyBag n) by SUBSET_1:def 1;
A3:   b*canFS(support b) = b*canFS({x}) by HILBASIS:8
      .= b*<*x*> by FINSEQ_1:94
      .= <*b.x*> by A1,FINSEQ_2:34;
      b.x in rng b by A1,FUNCT_1:3; then
reconsider f = b*canFS(support b) as FinSequence of NAT by A3,FINSEQ_1:74;
      degree b = Sum f by UPROOTS:def 4
      .= ((EmptyBag n)+*(x,1)).x by A3,HILBASIS:def 2
      .= 1 by A2,FUNCT_7:31;
      hence thesis;
    end;
