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 Th46:
  for L being add-associative right_zeroed right_complementable distributive
         non degenerated doubleLoopStr
  for p being monic Polynomial of L
  for q being Polynomial of L st deg p > deg q holds
  p-q is monic
  proof
    let L be add-associative right_zeroed right_complementable distributive
        non degenerated doubleLoopStr;
    let p be monic Polynomial of L;
    let q be Polynomial of L;
    assume
A1: deg p > deg q;
    then
A2: q.(len p-'1) = 0.L by Lm3;
    deg(p-q) = deg p by A1,Th41;
    hence LC(p-q) = p.(len p-'1) - q.(len p-'1) by NORMSP_1:def 3
    .= LC p by A2
    .= 1.L by RATFUNC1:def 7;
  end;
