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 Th47:
  for L being Abelian add-associative right_zeroed right_complementable
        unital distributive non empty doubleLoopStr
  for z1,z2 being Element of L
  for p being Polynomial of L st eval(p,z1) = z2
  holds eval(p-<%z2%>,z1) = 0.L
  proof
    let L be Abelian add-associative right_zeroed right_complementable
          unital distributive non empty doubleLoopStr;
    let z1,z2 be Element of L;
    let p be Polynomial of L such that
A1: eval(p,z1) = z2;
    thus eval(p-<%z2%>,z1) = eval(p,z1) - eval(<%z2%>,z1) by POLYNOM4:21
    .= z2 - z2 by A1,POLYNOM5:37
    .= 0.L by RLVECT_1:15;
  end;
