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 Th9:
    for a be Element of R,i be Element of n holds
    Support((1_1(i,R)) + a|(n,R)) c= {UnitBag i} \/ {EmptyBag n}
    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;
A1:   Support p = {UBi} by HILBASIS:13;
A2:   Support q = {} or Support q = {EmptyBag n} by POLYNOM7:22;
A5:   Support(p+q) c= Support p \/ Support q by POLYNOM1:20;
      Support q c= {EmptyBag n} by A2,XBOOLE_0:def 1; then
      Support p \/ Support q c= Support p \/ {EmptyBag n} by XBOOLE_1:9;
      hence thesis by A1,A5;
    end;
