
theorem Th39:
  for L being left_unital non empty doubleLoopStr for n being
  non zero Nat, i being Nat st i <> 0 & i <> n holds unital_poly(L,n).i = 0.L
proof
  let L be left_unital non empty doubleLoopStr, n be non zero Nat;
  let i be Nat such that
A1: i <> 0 and
A2: i <> n;
  set p = 0_.(L)+*(0,-(1_L));
A3: i in NAT by ORDINAL1:def 12;
  p+*(n,1_L).i = p.i by A2,FUNCT_7:32
    .= (0_.(L)).i by A1,FUNCT_7:32
    .= 0.L by A3,FUNCOP_1:7;
  hence thesis;
end;
