
theorem Th18:
  for L being add-associative right_zeroed right_complementable
  associative commutative well-unital distributive almost_left_invertible non
degenerated non empty doubleLoopStr, i being Integer, x being Element of L st
  x <> 0.L holds (pow(x, i))" = pow(x, -i)
proof
  let L be add-associative right_zeroed right_complementable associative
  commutative well-unital distributive almost_left_invertible non degenerated
  non empty doubleLoopStr;
  let i be Integer;
  let x be Element of L;
  assume
A1: x <> 0.L;
A2: 1.L <> 0.L;
  per cases;
  suppose
A3: i >= 0;
    per cases by A3,XREAL_1:24;
    suppose
A4:   - i < -0;
      hence pow(x, -i) = (pow(x, |.- i .|))" by Lm3
        .= (pow(x, (--i)))" by A4,ABSVALUE:def 1
        .= (pow(x, i))";
    end;
    suppose
A5:   i = 0;
      hence pow(x, (- i)) = 1.L by Th13
        .= 1.L * (1.L)" by A2,VECTSP_1:def 10
        .= (1.L)"
        .= (pow(x, i))" by A5,Th13;
    end;
  end;
  suppose
A6: i < 0;
A7: pow(x, |.i.|) = x |^ (|.i.|) by Def2;
    pow(x, i) = (pow(x, |.i.|))" by A6,Lm3;
    then (pow(x, i))" = pow(x, |.i.|) by A1,A7,Th1,VECTSP_1:24;
    hence thesis by A6,ABSVALUE:def 1;
  end;
end;
