
theorem Th36:
  for L be Abelian add-associative right_zeroed
  right_complementable well-unital associative commutative distributive
almost_left_invertible non empty doubleLoopStr for x be Element of L
  for n be Nat holds <%x%>`^n = <%power(x,n)%>
proof
  let L be Abelian add-associative right_zeroed right_complementable
  well-unital associative commutative distributive almost_left_invertible non
  empty doubleLoopStr;
  let x be Element of L;
  defpred P[Nat] means <%x%>`^$1 = <%power(x,$1)%>;
A1: for n be Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume <%x%>`^n = <%power(x,n)%>;
    hence <%x%>`^(n+1) = <%(power L).(x,n)%>*'<%x%> by Th19
      .= <%(power L).(x,n)*x%> by Th35
      .= <%power(x,n+1)%> by GROUP_1:def 7;
  end;
  <%x%>`^0 = 1_.(L) by Th15
    .= 1.L*1_.(L) by Th27
    .= <%1_L%> by Th29
    .= <%(power L).(x,0)%> by GROUP_1:def 7;
  then
A2: P[0];
  thus for n be Nat holds P[n] from NAT_1:sch 2(A2,A1);
end;
