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
    for a be Element of R,i be Element of n holds
    degree((1_1(i,R)) + a|(n,R)) = 1
    proof
      let a be Element of R,i be Element of n;
      set p = 1_1(i,R), q = a|(n,R);
      set UBi = UnitBag i;
      reconsider p0 = p + q as Polynomial of n,R;
A1:   Support p0 c= {UBi} \/ {EmptyBag n} by Th9;
      dom p0 = Bags n by FUNCT_2:def 1; then
A2:   UBi in dom p0 by PRE_POLY:def 12;
      p.UBi = 1.R & q.UBi = 0.R by Th6; then
      p0.UBi = 1.R + 0.R by POLYNOM1:15 .= 1.R; then
      p0.UBi <> 0.R; then
A4:   UBi in Support p0 by A2,POLYNOM1:def 3; then
      Support p0 <> {} by XBOOLE_0:def 1; then
A5:   p0 <> 0_(n,R) by Th1;
A6:   for o holds o in Support p0 implies o = UBi or o= EmptyBag n
      proof
        let o;
        assume o in Support p0; then
        o in {UBi} or o in {EmptyBag n} by A1,XBOOLE_0:def 3;
        hence thesis by TARSKI:def 1;
      end;
A8:   for s1 be bag of n st s1 in Support p0 holds
      degree s1 <= degree UBi
      proof
        let s1 be bag of n;
        assume s1 in Support p0; then
        per cases by A6;
          suppose s1 = UBi;
            hence thesis;
          end;
          suppose s1 = EmptyBag n;
            hence thesis by Th10;
          end;
        end;
       consider s be bag of n such that
A12:   s = UBi;
       degree p0 = degree s by A8,A5,A12,A4,HILB10_5:def 3;
       hence thesis by A12,Th11;
      end;
