 reserve R for domRing;
 reserve p for odd prime Nat, m for positive Nat;
 reserve g for non zero Polynomial of INT.Ring;

theorem Th43:
  for g be non zero Polynomial of INT.Ring holds
  ex M be Nat st for i be Nat holds |. g.i .| <= M
   proof
     let g be non zero Polynomial of INT.Ring;
reconsider g1 = |. g .| as Element of the carrier of Polynom-Ring INT.Ring
by POLYNOM3:def 10;
A1:  rng g1 = g1.:(Support g1) \/ {0.INT.Ring} by E_TRANS1:8;
     rng |. g .| c= NAT
     proof
       let o be object;
       assume o in rng |. g .|; then
       consider x be object such that
A3:    x in dom |. g .| & (|. g .|).x = o by FUNCT_1:def 3;
       (|. g .|).x = |. g.x .| by A3,Def9;
       hence thesis by A3;
     end; then
reconsider S = rng |. g .| as finite non empty natural-membered set by A1;
     max S in S by XXREAL_2:def 8; then
     reconsider M = max S as Nat;
     take M;
A5:  dom |. g .| = NAT by FUNCT_2:def 1;
     for i be Nat holds |. g.i .| <= M
     proof
       let i be Nat;
       (|. g .|).i = |. g.i .| by Def9; then
       |. g.i .| in S by A5,ORDINAL1:def 12,FUNCT_1:3;
       hence thesis by XXREAL_2:def 8;
     end;
     hence thesis;
   end;
