
theorem Th43:
  for n being Ordinal, T being connected admissible TermOrder of n
  , L being add-associative right_zeroed right_complementable non trivial
  addLoopStr, p being Polynomial of n,L, i being Element of NAT st i < card(
  Support p) holds Low(p,T,i+1) = Red(Low(p,T,i),T)
proof
  let n be Ordinal, T be connected admissible TermOrder of n, L be
  add-associative right_zeroed right_complementable non trivial addLoopStr, p
  be Polynomial of n,L, i be Element of NAT;
  set l = Low(p,T,i), l1 = Low(p,T,i+1), r = Red(l,T);
  assume
A1: i < card(Support p);
  then
A2: Support(Low(p,T,i)) c= Support(p) by Th26;
  Support Low(p,T,i) = Lower_Support(p,T,i) by A1,Lm3;
  then
A3: card Support l = card(Support p) - i by A1,Th24;
A4: Support(Low(p,T,i+1)) c= Support(Low(p,T,i)) by A1,Th41;
A5: i + 1 <= card(Support p) by A1,NAT_1:13;
  then Support Low(p,T,i+1) = Lower_Support(p,T,i+1) by Lm3;
  then
A6: card Support l1 = card(Support p) - (i+1) by A5,Th24;
A7: Support Low(p,T,i+1) = Lower_Support(p,T,i+1) by A5,Lm3;
  now
    set u = the Element of {HT(l,T)} /\ Support l1;
    assume
A8: {HT(l,T)} /\ Support l1 <> {};
    then u in {HT(l,T)} by XBOOLE_0:def 4;
    then
A9: u = HT(l,T) by TARSKI:def 1;
A10: u in Support l1 by A8,XBOOLE_0:def 4;
    now
      let u9 be object;
      assume
A11:  u9 in Support l;
      then reconsider u = u9 as Element of Bags n;
      u <= HT(Low(p,T,i),T),T by A11,TERMORD:def 6;
      hence u9 in Support l1 by A5,A2,A7,A10,A9,A11,Th24;
    end;
    then Support l c= Support l1;
    then card(Support p) + -i <= card(Support p) + -(i+1) by A3,A6,NAT_1:43;
    then - i <= -(i + 1) by XREAL_1:6;
    then i + 1 <= i by XREAL_1:24;
    then (i + 1) - i <= i - i by XREAL_1:9;
    then 1 <= 0;
    hence contradiction;
  end;
  then
A12: Support l1 misses {HT(l,T)} by XBOOLE_0:def 7;
A13: Support Low(p,T,i) \ Support Low(p,T,i+1) = {HT(Low(p,T,i),T)} by A1,Th42;
  then Support Low(p,T,i) = Support l1 \/ {HT(l,T)} by A1,Th41,XBOOLE_1:45;
  then
A14: Support r = (Support l1 \/ {HT(l,T)}) \ {HT(l,T)} by TERMORD:36
    .= Support l1 \ {HT(l,T)} by XBOOLE_1:40
    .= Support l1 by A12,XBOOLE_1:83;
A15: now
    let x be object;
    assume x in dom l1;
    then reconsider b = x as Element of Bags n;
    now
      per cases;
      case
A16:    b in Support l1;
        then not b in {HT(Low(p,T,i),T)} by A13,XBOOLE_0:def 5;
        then
A17:    b <> HT(l,T) by TARSKI:def 1;
        thus l1.b = p.b by A5,A16,Th31
          .= l.b by A1,A4,A16,Th31
          .= r.b by A4,A16,A17,TERMORD:40;
      end;
      case
A18:    not b in Support l1;
        hence l1.b = 0.L by POLYNOM1:def 4
          .= r.b by A14,A18,POLYNOM1:def 4;
      end;
    end;
    hence l1.x = r.x;
  end;
  dom l1 = Bags n by FUNCT_2:def 1
    .= dom r by FUNCT_2:def 1;
  hence thesis by A15,FUNCT_1:2;
end;
