
theorem
  for L be Abelian add-associative right_zeroed right_complementable
     well-unital commutative distributive non empty doubleLoopStr
  for n be Nat holds 1_.(L)`^n = 1_.(L)
proof
  let L be Abelian add-associative right_zeroed right_complementable
  well-unital commutative distributive non empty doubleLoopStr;
  defpred P[Nat] means 1_.(L)`^$1 = 1_.(L);
A1: now
    let n be Nat;
    assume P[n];
    then 1_.(L)`^(n+1) = (1_.(L))*'1_.(L) by Th19
      .= 1_.(L) by POLYNOM3:35;
    hence P[n+1];
  end;
A2: P[0] by Th15;
  thus for n be Nat holds P[n] from NAT_1:sch 2(A2,A1);
end;
