
theorem Th56:
  for n being Ordinal, T being connected admissible TermOrder of n
  , L being add-associative right_complementable right_zeroed commutative
  associative well-unital distributive Abelian almost_left_invertible non
  trivial doubleLoopStr, p1,p2 being Polynomial of n,L st HT(p1,T),HT(p2,T)
  are_disjoint holds PolyRedRel({p1,p2},T) reduces S-Poly(p1,p2,T),0_(n,L)
proof
  let n be Ordinal, T be connected admissible TermOrder of n, L be
  add-associative right_complementable right_zeroed commutative associative
  well-unital distributive Abelian almost_left_invertible non trivial
  doubleLoopStr, p1,p2 be Polynomial of n,L;
  assume
A1: HT(p1,T),HT(p2,T) are_disjoint;
  then
A2: S-Poly(p1,p2,T) = Red(p1,T) *' p2 - Red(p2,T) *' p1 by Th54;
  now
    per cases;
    case
      p1 = 0_(n,L);
      then Red(p2,T) *' p1 = 0_(n,L) & Red(p1,T) = 0_(n,L) by Th11,POLYNOM1:28;
      then S-Poly(p1,p2,T) = 0_(n,L) - 0_(n,L) by A2,POLYNOM1:28
        .= 0_(n,L) by POLYRED:4;
      hence thesis by REWRITE1:12;
    end;
    case
      p1 <> 0_(n,L);
      then reconsider p1a = p1 as non-zero Polynomial of n,L by POLYNOM7:def 1;
      now
        let u be object;
        assume u in {p2};
        then u = p2 by TARSKI:def 1;
        hence u in {p1,p2} by TARSKI:def 2;
      end;
      then
A3:   {p2} c= {p1,p2};
      then
A4:   PolyRedRel({p2},T) c= PolyRedRel({p1,p2},T) by GROEB_1:4;
      now
        let u be object;
        assume u in {p1};
        then u = p1 by TARSKI:def 1;
        hence u in {p1,p2} by TARSKI:def 2;
      end;
      then
A5:   {p1} c= {p1,p2};
      then
A6:   PolyRedRel({p1},T) c= PolyRedRel({p1,p2},T) by GROEB_1:4;
      now
        per cases;
        case
          p2 = 0_(n,L);
          then Red(p1,T) *' p2 = 0_(n,L) & Red(p2,T) = 0_(n,L) by Th11,
POLYNOM1:28;
          then S-Poly(p1,p2,T) = 0_(n,L) - 0_(n,L) by A2,POLYNOM1:28
            .= 0_(n,L) by POLYRED:4;
          hence thesis by REWRITE1:12;
        end;
        case
          p2 <> 0_(n,L);
          then reconsider p2a = p2 as non-zero Polynomial of n,L by
POLYNOM7:def 1;
          now
            per cases;
            case
              Red(p1,T) = 0_(n,L);
              then
A7:           S-Poly(p1,p2,T) = 0_(n,L) - Red(p2,T) *' p1 by A2,POLYNOM1:28;
              now
                per cases;
                case
                  Red(p2,T) = 0_(n,L);
                  then S-Poly(p1,p2,T) = 0_(n,L) - 0_(n,L) by A7,POLYNOM1:28
                    .= 0_(n,L) by POLYRED:4;
                  hence thesis by REWRITE1:12;
                end;
                case
                  Red(p2,T) <> 0_(n,L);
                  then reconsider
                  rp2 = Red(p2,T) as non-zero Polynomial of n,L by
POLYNOM7:def 1;
                  PolyRedRel({p1a},T) reduces -(rp2*'p1a),-0_(n,L) by Th49,Th51
;
                  then PolyRedRel({p1a},T) reduces -(rp2*'p1a),0_(n,L) by Th13;
                  then PolyRedRel({p1},T) reduces S-Poly(p1,p2,T),0_(n,L) by A7
,Th14;
                  hence thesis by A5,GROEB_1:4,REWRITE1:22;
                end;
              end;
              hence thesis;
            end;
            case
              Red(p1,T) <> 0_(n,L);
              then reconsider
              rp1 = Red(p1,T) as non-zero Polynomial of n,L by POLYNOM7:def 1;
              now
                per cases;
                case
                  Red(p2,T) = 0_(n,L);
                  then Red(p2,T) *' p1 = 0_(n,L) by POLYNOM1:28;
                  then
A8:               S-Poly(p1,p2,T) = Red(p1,T) *' p2 - 0_ (n,L) by A1,Th54
                    .= Red(p1,T) *' p2 by POLYRED:4;
                  PolyRedRel({p2a},T) reduces rp1*'p2a,0_(n,L) by Th51;
                  hence thesis by A3,A8,GROEB_1:4,REWRITE1:22;
                end;
                case
                  Red(p2,T) <> 0_(n,L);
                  then reconsider
                  rp2 = Red(p2,T) as non-zero Polynomial of n,L by
POLYNOM7:def 1;
                  S-Poly(p1,p2,T) = HM(p2a,T)*'rp1 - HM(p1a,T)*'rp2 by A1,Th53;
                  then
A9:               PolyRedRel({p1,p2},T) reduces S-Poly(p1,p2,T),p2 *' Red
                  (p1,T) by A1,A6,Th55,REWRITE1:22;
                  PolyRedRel({p1,p2},T) reduces rp1*'p2a,0_(n,L) by A4,Th51,
REWRITE1:22;
                  hence thesis by A9,REWRITE1:16;
                end;
              end;
              hence thesis;
            end;
          end;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
