
theorem LT1:
for R being non degenerated Ring
for n being Ordinal
for p being Polynomial of n,R
for b being bag of n st b in Support p
holds b = Lt p iff for b1 being bag of n st b1 in Support p holds b1 <=' b
proof
let R be non degenerated Ring, n be Ordinal, p be Polynomial of n,R;
let b be bag of n;
set F = SgmX(BagOrder n,Support p);
assume AS1: b in Support p; then
AS2: p <> 0_(n,R) by YY;
     field(BagOrder n) = Bags n by ORDERS_1:12; then
AS3: BagOrder n linearly_orders Support p by ORDERS_1:37,ORDERS_1:38;
     card(Support p) >= 0 + 1 by AS1,INT_1:7; then
     1 <= len F by AS3,PRE_POLY:11; then
     len F in Seg(len F); then
AS4: len F in dom F by FINSEQ_1:def 3; then
AS5: F.(len F) = F/.(len F) by PARTFUN1:def 6;
A: now assume AS6: b = Lt p; then
   A0: b = F.(len F) by AS2,defLT;
   thus for b1 being bag of n st b1 in Support p holds b1 <=' b
     proof
     let b1 be bag of n;
     assume b1 in Support p; then
     b1 in rng F by AS3,PRE_POLY:def 2; then
     consider o being object such that
     A1: o in dom F & b1 = F.o by FUNCT_1:def 3;
     reconsider i = o as Element of NAT by A1;
     A2: F.i = F/.i by A1,PARTFUN1:def 6;
     i in Seg(len F) by A1,FINSEQ_1:def 3; then
     i <= len F by FINSEQ_1:1; then
     per cases by XXREAL_0:1;
     suppose i = len F;
       hence thesis by AS6,A1,AS2,defLT;
       end;
     suppose i < len F;
       then [b1,b] in BagOrder n by A0,A1,A2,AS4,AS5,AS3,PRE_POLY:def 2;
       hence b1 <=' b by PRE_POLY:def 14;
       end;
     end;
   end;
now assume B: for b1 being bag of n st b1 in Support p holds b1 <=' b;
   now let y being Element of Bags n;
     assume y in Support p;
     then y <=' b by B;
     hence [y,b] in BagOrder n by PRE_POLY:def 14;
     end;
   then b = F/.(len F) by AS1,AS3,PRE_POLY:21
         .= F.(len F) by AS4,PARTFUN1:def 6;
   hence b = Lt p by AS2,defLT;
   end;
hence thesis by A;
end;
