
theorem Th33:
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_zeroed right_complementable non empty addLoopStr, p
  being Polynomial of n,L, i being Element of NAT st i <= card(Support p) holds
  Up(p,T,i) + Low(p,T,i) = p
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
right_zeroed right_complementable non empty addLoopStr, p be Polynomial of n,
  L, i be Element of NAT;
  set u = Up(p,T,i) + Low(p,T,i);
  assume
A1: i <= card(Support p);
A2: now
    let x be object;
    assume
A3: x in Support p;
    then reconsider x9 = x as Element of Bags n;
A4: u.x9 = Up(p,T,i).x9 + Low(p,T,i).x9 by POLYNOM1:15;
A5: now
      per cases by A1,A3,Th28;
      case
A6:     x9 in Support Up(p,T,i);
        hence u.x9 = Up(p,T,i).x9 + 0.L by A1,A4,Th32
          .= Up(p,T,i).x9 by RLVECT_1:def 4
          .= p.x9 by A1,A6,Th32;
      end;
      case
A7:     x9 in Support Low(p,T,i);
        hence u.x9 = 0.L + Low(p,T,i).x9 by A1,A4,Th31
          .= Low(p,T,i).x9 by ALGSTR_1:def 2
          .= p.x9 by A1,A7,Th31;
      end;
    end;
    p.x9 <> 0.L by A3,POLYNOM1:def 4;
    hence x in Support u by A5,POLYNOM1:def 4;
  end;
  now
    let x be object;
    Support Up(p,T,i) c= Support p & Support Low(p,T,i) c= Support p by A1,Th26
;
    then
    Support u c= Support Up(p,T,i) \/ Support Low(p,T,i) & Support Up(p,T,
    i) \/ Support Low(p,T,i) c= Support p by POLYNOM1:20,XBOOLE_1:8;
    then
A8: Support u c= Support p;
    assume x in Support u;
    hence x in Support p by A8;
  end;
  then
A9: Support u = Support p by A2,TARSKI:2;
A10: now
    let x be object;
    assume x in dom p;
    then reconsider x9 = x as Element of Bags n;
A11: u.x9 = Up(p,T,i).x9 + Low(p,T,i).x9 by POLYNOM1:15;
    now
      per cases;
      case
A12:    x9 in Support p;
        now
          per cases by A1,A12,Th28;
          case
A13:        x9 in Support Up(p,T,i);
            hence u.x9 = Up(p,T,i).x9 + 0.L by A1,A11,Th32
              .= Up(p,T,i).x9 by RLVECT_1:def 4
              .= p.x9 by A1,A13,Th32;
          end;
          case
A14:        x9 in Support Low(p,T,i);
            hence u.x9 = 0.L + Low(p,T,i).x9 by A1,A11,Th31
              .= Low(p,T,i).x9 by ALGSTR_1:def 2
              .= p.x9 by A1,A14,Th31;
          end;
        end;
        hence p.x9 = u.x9;
      end;
      case
A15:    not x9 in Support p;
        hence p.x9 = 0.L by POLYNOM1:def 4
          .= u.x9 by A9,A15,POLYNOM1:def 4;
      end;
    end;
    hence p.x = u.x;
  end;
  dom p = Bags n by FUNCT_2:def 1
    .= dom u by FUNCT_2:def 1;
  hence thesis by A10,FUNCT_1:2;
end;
