
theorem Th38:
  for L being left_unital non empty doubleLoopStr for n being
non zero Element of NAT holds unital_poly(L,n).0 = -1_L & unital_poly(L,n).n =
  1_L
proof
  let L be left_unital non empty doubleLoopStr, n be non zero Element of
  NAT;
  set p = 0_.(L)+*(0,-(1_L));
  set q = 0_.(L)+*(n,1_L);
  0 in NAT;
  then
A1: unital_poly(L,n) = q+*(0,-(1_L)) & 0 in dom q by FUNCT_7:33,NORMSP_1:12;
  n in NAT;
  then n in dom p by NORMSP_1:12;
  hence thesis by A1,FUNCT_7:31;
end;
