
theorem Th42:
  for n being Ordinal, T being connected admissible TermOrder of n
  , L being add-associative right_complementable right_zeroed commutative
  associative well-unital distributive almost_left_invertible non degenerated
non empty doubleLoopStr, f,p,g being Polynomial of n,L holds f reduces_to g,p,
  T implies for b being bag of n st b in Support g holds b <= HT(f,T),T
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 almost_left_invertible non degenerated non empty
  doubleLoopStr, f,p,g be Polynomial of n,L;
A1: T is_connected_in field T by RELAT_2:def 14;
  assume f reduces_to g,p,T;
  then consider b being bag of n such that
A2: f reduces_to g,p,b,T;
  b in Support f by A2;
  then
A3: b <= HT(f,T),T by TERMORD:def 6;
  now
    let u be bag of n;
    assume
A4: u in Support g;
    now
      per cases;
      case
        u = b;
        hence contradiction by A2,A4,Lm17;
      end;
      case
A5:     u <> b;
        b <= b,T by TERMORD:6;
        then [b,b] in T by TERMORD:def 2;
        then
A6:     b in field T by RELAT_1:15;
        u <= u,T by TERMORD:6;
        then [u,u] in T by TERMORD:def 2;
        then u in field T by RELAT_1:15;
        then
A7:     [u,b] in T or [b,u] in T by A1,A5,A6,RELAT_2:def 6;
        now
          per cases by A7,TERMORD:def 2;
          case
            u <= b,T;
            hence u <= HT(f,T),T by A3,TERMORD:8;
          end;
          case
            b <= u,T;
            then b < u,T by A5,TERMORD:def 3;
            then u in Support f iff u in Support g by A2,Th40;
            hence u <= HT(f,T),T by A4,TERMORD:def 6;
          end;
        end;
        hence u <= HT(f,T),T;
      end;
    end;
    hence u <= HT(f,T),T;
  end;
  hence thesis;
end;
