
theorem Th16:
  for L being associative commutative well-unital distributive
  almost_left_invertible non degenerated non empty doubleLoopStr, i being
  Integer holds pow(1.L,i) = 1.L
proof
  let L be associative commutative well-unital distributive
  almost_left_invertible non degenerated non empty doubleLoopStr;
  let i be Integer;
  per cases;
  suppose
    0 <= i;
    then i is Element of NAT by INT_1:3;
    hence thesis by Lm5;
  end;
  suppose
A1: 0 > i;
A2: 1.L <> 0.L & 1.L * 1.L = 1.L;
A3: pow(1.L,|.i.|) = 1.L by Lm5;
    pow(1.L,i) = (power(L).(1.L,|.i.|))" by A1,Def2
      .= (1.L)" by A3,Def2;
    hence thesis by A2,VECTSP_1:def 10;
  end;
end;
