
theorem Th8:
  for n being Ordinal, T being admissible connected TermOrder of n,
  L being add-associative right_complementable right_zeroed commutative
  associative well-unital distributive almost_left_invertible non trivial
  doubleLoopStr, p,q being Polynomial of n,L holds Support(q) c= Support(p)
  implies q <= p,T
proof
  let n be Ordinal, T be admissible connected TermOrder of n, L be
  add-associative right_complementable right_zeroed commutative associative
well-unital distributive almost_left_invertible non trivial doubleLoopStr, p,
  q be Polynomial of n,L;
  assume
A1: Support q c= Support p;
  defpred P[Nat] means for f,g being Polynomial of n,L st Support f
  c= Support g & card(Support f) = $1 holds f <= g,T;
A2: now
    let k be Nat;
    assume
A3: P[k];
    now
      set R = RelStr(# Bags n, T#);
      let f,g be Polynomial of n,L;
      assume that
A4:   Support f c= Support g and
A5:   card(Support f) = k+1;
      set rf = Red(f,T), rg = Red(g,T);
A6:   Support f <> {} by A5;
      then
A7:   HT(f,T) in Support f by TERMORD:def 6;
      f <> 0_(n,L) by A6,POLYNOM7:1;
      then
A8:   f is non-zero by POLYNOM7:def 1;
      g <> 0_(n,L) by A4,A7,POLYNOM7:1;
      then
A9:   g is non-zero by POLYNOM7:def 1;
      now
        per cases;
        case
A10:      HT(f,T) = HT(g,T);
A11:      Support(rf) = Support(f) \ {HT(f,T)} by TERMORD:36;
A12:      Support(rg) = Support(g) \ {HT(g,T)} by TERMORD:36;
          now
            let u be object;
            assume u in Support(rf);
            then u in Support f & not u in {HT(f,T)} by A11,XBOOLE_0:def 5;
            hence u in Support(rg) by A4,A10,A12,XBOOLE_0:def 5;
          end;
          then
A13:      Support(rf) c= Support(rg);
          for u being object holds u in {HT(f,T)} implies u in Support f
    by A7,
TARSKI:def 1;
          then
A14:      {HT(f,T)} c= Support f;
A15:      Support(f,T) <> {} & Support(g,T) <> {} by A4,A7,POLYRED:def 4;
A16:      Support(rf,T) = Support rf by POLYRED:def 4;
          HT(f,T) in {HT(f,T)} by TARSKI:def 1;
          then
A17:      not HT(f,T) in Support Red(f,T) by A11,XBOOLE_0:def 5;
A18:      PosetMax(Support(f,T)) = HT(g,T) by A8,A10,POLYRED:24
            .= PosetMax(Support(g,T)) by A9,POLYRED:24;
A19:      Support(rg,T) = Support rg by POLYRED:def 4;
A20:      Support(g,T) = Support g by POLYRED:def 4;
          then
A21:      Support(g,T)\{PosetMax(Support(g,T))} = Support(rg,T) by A9,A12,A19,
POLYRED:24;
          Support(rf) \/ {HT(f,T)} = Support f \/ {HT(f,T)} by A11,XBOOLE_1:39
            .= Support f by A14,XBOOLE_1:12;
          then card(Support(Red(f,T))) + 1 = k + 1 by A5,A17,CARD_2:41;
          then rf <= rg,T by A3,A13;
          then [Support rf,Support rg] in FinOrd RelStr(# Bags n, T#) by
POLYRED:def 2;
          then
A22:      [Support(rf,T),Support(rg,T)] in union rng FinOrd-Approx R by A16,A19
,BAGORDER:def 15;
A23:      Support(f,T) = Support f by POLYRED:def 4;
          then Support(f,T)\{PosetMax(Support(f,T))} = Support(rf,T) by A8,A11
,A16,POLYRED:24;
          then
          [Support(f,T),Support(g,T)] in union rng FinOrd-Approx R by A22,A15
,A18,A21,BAGORDER:35;
          then
          [Support f,Support g] in FinOrd RelStr(# Bags n, T#) by A23,A20,
BAGORDER:def 15;
          hence f <= g,T by POLYRED:def 2;
        end;
        case
A24:      HT(f,T) <> HT(g,T);
          now
            assume HT(g,T) < HT(f,T),T;
            then not HT(f,T) <= HT(g,T),T by TERMORD:5;
            hence contradiction by A4,A7,TERMORD:def 6;
          end;
          then HT(f,T) <= HT(g,T),T by TERMORD:5;
          then HT(f,T) < HT(g,T),T by A24,TERMORD:def 3;
          then f < g,T by POLYRED:32;
          hence f <= g,T by POLYRED:def 3;
        end;
      end;
      hence f <= g,T;
    end;
    hence P[k+1];
  end;
A25: ex k being Element of NAT st card(Support q) = k;
A26: P[ 0 ]
  proof
    let f,g be Polynomial of n,L;
    assume that
    Support f c= Support g and
A27: card(Support f) = 0;
    Support f = {} by A27;
    then f = 0_(n,L) by POLYNOM7:1;
    hence thesis by POLYRED:30;
  end;
  for k being Nat holds P[k] from NAT_1:sch 2(A26,A2);
  hence thesis by A1,A25;
end;
