
theorem Th25:
  for L being Field, x being Element of L st x <> 0.L for i being
  Integer holds pow(x", i) = (pow(x, i))"
proof
  let L be Field;
  let x be Element of L;
  assume
A1: x <> 0.L;
  let i be Integer;
  per cases;
  suppose
    i >= 0;
    then reconsider n = i as Element of NAT by INT_1:3;
    thus pow(x", i) = (pow(x, n))" by A1,Lm9
      .= (pow(x, i))";
  end;
  suppose
A2: i < 0;
A3: pow(x, |.i.|) = x |^ (|.i.|) by Def2;
    thus pow(x", i) = (pow(x", |.i.|))" by A2,Lm3
      .= pow((x")", |.i.|) by A1,Lm9,VECTSP_1:25
      .= pow(x, |.i.|) by A1,VECTSP_1:24
      .= (pow(x, |.i.|))"" by A1,A3,Th1,VECTSP_1:24
      .= (pow(x, i))" by A2,Lm3;
  end;
end;
