
theorem
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_complementable right_zeroed non trivial addLoopStr, p
  being Polynomial of n,L holds Support(HM(p,T) + Red(p,T)) = Support p
proof
  let n being Ordinal, O being connected TermOrder of n, L being
  add-associative right_complementable right_zeroed non trivial addLoopStr, p
  being Polynomial of n,L;
A1: now
    let u be object;
    assume
A2: u in Support p;
    then reconsider u9 = u as Element of Bags n;
    reconsider u9 as bag of n;
A3: p.u9 <> 0.L by A2,POLYNOM1:def 4;
    now
      per cases;
      case
A4:     u9 = HT(p,O);
        then
A5:     p.(HT(p,O)) <> 0.L by A2,POLYNOM1:def 4;
        (HM(p,O) + Red(p,O)).u9 = HM(p,O).u9 + Red(p,O).u9 by POLYNOM1:15
          .= HM(p,O).u9 + 0.L by A4,Lm18
          .= HM(p,O).u9 by RLVECT_1:4
          .= HC(p,O) by A4,Lm8;
        hence u9 in Support(HM(p,O) + Red(p,O)) by A5,POLYNOM1:def 4;
      end;
      case
A6:     u9 <> HT(p,O);
        (HM(p,O) + Red(p,O)).u9 = HM(p,O).u9 + Red(p,O).u9 by POLYNOM1:15
          .= HM(p,O).u9 + p.u9 by A6,Lm19
          .= 0.L + p.u9 by A6,Th19
          .= p.u9 by RLVECT_1:4;
        hence u9 in Support(HM(p,O) + Red(p,O)) by A3,POLYNOM1:def 4;
      end;
    end;
    hence u in Support(HM(p,O) + Red(p,O));
  end;
  now
    let u be object;
    assume
A7: u in Support(HM(p,O) + Red(p,O));
    then reconsider u9 = u as Element of Bags n;
    reconsider u9 as bag of n;
A8: (HM(p,O) + Red(p,O)).u9 <> 0.L by A7,POLYNOM1:def 4;
    now
      per cases;
      case
A9:     u9 = HT(p,O);
        (HM(p,O) + Red(p,O)).u9 = HM(p,O).u9 + Red(p,O).u9 by POLYNOM1:15
          .= HM(p,O).HT(p,O) + 0.L by A9,Lm18
          .= HM(p,O).HT(p,O) by RLVECT_1:4
          .= p.u9 by A9,Lm8;
        hence u9 in Support p by A8,POLYNOM1:def 4;
      end;
      case
A10:    u9 <> HT(p,O);
        (HM(p,O) + Red(p,O)).u9 = HM(p,O).u9 + Red(p,O).u9 by POLYNOM1:15
          .= 0.L + Red(p,O).u9 by A10,Th19
          .= Red(p,O).u9 by RLVECT_1:4
          .= p.u by A10,Lm19;
        hence u9 in Support p by A8,POLYNOM1:def 4;
      end;
    end;
    hence u in Support p;
  end;
  hence thesis by A1,TARSKI:2;
end;
