reserve r,t for Real;
reserve i for Integer;
reserve k,n for Nat;
reserve p for Polynomial of F_Real;
reserve e for Element of F_Real;
reserve L for non empty ZeroStr;
reserve z,z0,z1,z2 for Element of L;

theorem Th41:
  for L being add-associative right_zeroed right_complementable
      non empty addLoopStr
  for p,q being Polynomial of L st deg p > deg q holds
  deg(p-q) = deg p
  proof
    let L be add-associative right_zeroed right_complementable
        non empty addLoopStr;
    let p,q be Polynomial of L;
    assume
A1: deg p > deg q;
A2: deg q = deg(-q) by POLYNOM4:8;
    then deg(p+-q) = max(deg(p),deg(-q)) by A1,HURWITZ:21;
    hence thesis by A1,A2,XXREAL_0:def 10;
  end;
