
theorem Th44:
  for n being Ordinal, T being connected admissible TermOrder of n
  , L being add-associative right_complementable right_zeroed well-unital
distributive domRing-like non trivial doubleLoopStr, p being Polynomial of n,
  L, m being non-zero Monomial of n,L, i being Element of NAT st i <= card(
  Support p) holds Low(m*'p,T,i) = m *' Low(p,T,i)
proof
  let n be Ordinal, T be connected admissible TermOrder of n, L be
  add-associative right_complementable right_zeroed well-unital distributive
domRing-like non trivial doubleLoopStr, p be Polynomial of n,L, m be non-zero
  Monomial of n,L, i be Element of NAT;
  set l = Low(p,T,i), lm = Low(m*'p,T,i);
  assume
A1: i <= card(Support p);
  then
A2: i <= card(Support(m*'p)) by Th10;
A3: Support(m*'l) c= {s + t where s,t is Element of Bags n : s in Support m
  & t in Support l} by TERMORD:30;
A4: now
    m <> 0_(n,L) by POLYNOM7:def 1;
    then Support m <> {} by POLYNOM7:1;
    then
A5: Support m = {term(m)} by POLYNOM7:7;
    then term(m) in Support m by TARSKI:def 1;
    then
A6: m.term(m) <> 0.L by POLYNOM1:def 4;
    let u be object;
    assume
A7: u in Support m*'l;
    then reconsider u9 = u as Element of Bags n;
    u in {s + t where s,t is Element of Bags n : s in Support m & t in
    Support l} by A3,A7;
    then consider s,t being Element of Bags n such that
A8: u9 = s + t and
A9: s in Support m and
A10: t in Support l;
A11: l.t <> 0.L by A10,POLYNOM1:def 4;
A12: term(m) = s by A9,A5,TARSKI:def 1;
    then (m*'p).u9 = m.term(m) * p.t by A8,POLYRED:7
      .= m.term(m) * l.t by A1,A10,Th31;
    then (m*'p).u9 <> 0.L by A11,A6,VECTSP_2:def 1;
    then
A13: u9 in Support(m*'p) by POLYNOM1:def 4;
    now
      assume not s+t in Support Low(m*'p,T,i);
      then
A14:  s+t in Support Up(m*'p,T,i) by A2,A8,A13,Th28;
      now
        let t9 be bag of n;
        assume t9 in Support Low(p,T,i);
        then s+t9 in Support Low(m*'p,T,i) by A1,A12,Th40;
        then
A15:    s+t9 < s+t,T by A2,A14,Th29;
        not t <= t9,T by A15,TERMORD:5,Th2;
        hence t9 < t,T by TERMORD:5;
      end;
      then t < t,T by A10;
      hence contradiction by TERMORD:def 3;
    end;
    hence u in Support lm by A8;
  end;
A16: Support(m*'p) c= {s + t where s,t is Element of Bags n : s in Support m
  & t in Support p} by TERMORD:30;
  now
    let u be object;
    assume
A17: u in Support Low(m*'p,T,i);
    then reconsider u9 = u as Element of Bags n;
    Support Low(m*'p,T,i) c= Support(m*'p) by A2,Th26;
    then u9 in Support(m*'p) by A17;
    then
A18: u9 in {s + t where s,t is Element of Bags n : s in Support m & t in
    Support p} by A16;
    m <> 0_(n,L) by POLYNOM7:def 1;
    then Support m <> {} by POLYNOM7:1;
    then
A19: Support m = {term(m)} by POLYNOM7:7;
    then term(m) in Support m by TARSKI:def 1;
    then
A20: m.term(m) <> 0.L by POLYNOM1:def 4;
    consider s,t being Element of Bags n such that
A21: u = s + t and
A22: s in Support m and
A23: t in Support p by A18;
A24: p.t <> 0.L by A23,POLYNOM1:def 4;
A25: term(m) = s by A22,A19,TARSKI:def 1;
    then
A26: t in Support l by A1,A17,A21,Th40;
    (m*'l).(term(m)+t) = m.term(m) * l.t by POLYRED:7
      .= m.term(m) * p.t by A1,A26,Th31;
    then (m*'l).u9 <> 0.L by A21,A20,A25,A24,VECTSP_2:def 1;
    hence u in Support(m*'Low(p,T,i)) by POLYNOM1:def 4;
  end;
  then
A27: Support(m*'l) = Support lm by A4,TARSKI:2;
A28: now
    let x be object;
    assume x in dom(m*'l);
    then reconsider b = x as Element of Bags n;
    now
      per cases;
      case
A29:    b in Support(m*'l);
        then
A30:    b in {s + t where s,t is Element of Bags n : s in Support m & t
        in Support l} by A3;
A31:    b in Support lm by A4,A29;
        consider s,t being Element of Bags n such that
A32:    b = s + t and
A33:    s in Support m and
A34:    t in Support l by A30;
        Support m = {term(m)} by A33,POLYNOM7:7;
        then
A35:    term(m) = s by A33,TARSKI:def 1;
        hence (m*'l).b = m.term(m) * l.t by A32,POLYRED:7
          .= m.term(m) * p.t by A1,A34,Th31
          .= (m*'p).b by A32,A35,POLYRED:7
          .= lm.b by A2,A31,Th31;
      end;
      case
A36:    not b in Support(m*'l);
        hence (m*'l).b = 0.L by POLYNOM1:def 4
          .= lm.b by A27,A36,POLYNOM1:def 4;
      end;
    end;
    hence (m*'l).x = lm.x;
  end;
  dom(m*'l) = Bags n by FUNCT_2:def 1
    .= dom lm by FUNCT_2:def 1;
  hence thesis by A28,FUNCT_1:2;
end;
