
theorem Th19:
  for n being Ordinal, L being add-associative right_zeroed
  right_complementable distributive non trivial doubleLoopStr, p being
  Polynomial of n,L, a being Element of L holds Support(a*p) c= Support(p)
proof
  let n be Ordinal, L be add-associative right_zeroed right_complementable
distributive non trivial doubleLoopStr, p be Polynomial of n,L, a9 be Element
  of L;
A1: dom(0_(n,L)) = Bags n by FUNCT_2:def 1
    .= dom(a9*p) by FUNCT_2:def 1;
  per cases;
  suppose
A2: a9 = 0.L;
    now
      let u be object;
      assume u in dom(a9*p);
      then reconsider u9 = u as Element of Bags n;
      (a9*p).u9 = a9 * p.u9 by POLYNOM7:def 9
        .= 0.L by A2
        .= (0_(n,L)).u9 by POLYNOM1:22;
      hence (a9*p).u = (0_(n,L)).u;
    end;
    then a9*p = 0_(n,L) by A1,FUNCT_1:2;
    then for u being object holds u in Support(a9*p) implies u in Support(p)
       by POLYNOM7:1;
    hence thesis by TARSKI:def 3;
  end;
  suppose
    a9 <> 0.L;
    then reconsider a = a9 as non zero Element of L by STRUCT_0:def 12;
    now
      let u be object;
      assume
A3:   u in Support(a*p);
      then reconsider u9 = u as Element of Bags n;
A4:   (a*p).u9 = a * p.u9 by POLYNOM7:def 9;
      (a*p).u9 <> 0.L by A3,POLYNOM1:def 4;
      then p.u9 <> 0.L by A4;
      hence u in Support(p) by POLYNOM1:def 4;
    end;
    hence thesis by TARSKI:def 3;
  end;
end;
