reserve x, y, z, s for ExtReal;
reserve i, j for Integer;
reserve n, m for Nat;
reserve x, y, v, u for ExtInt;
reserve
  D for non empty doubleLoopStr,
  A for Subset of D;
reserve K for Field-like non degenerated
  associative add-associative right_zeroed right_complementable
  distributive Abelian non empty doubleLoopStr,
  a, b, c for Element of K;

theorem Th16:
  a <> 0.K implies a|^i <> 0.K
  proof
    assume
A1: a <> 0.K;
    per cases;
    suppose 0 <= i;
      then reconsider n1 = i as Element of NAT by INT_1:3;
      a |^ i = a |^ n1;
      hence a |^ i <> 0.K by A1,Lm3;
    end;
    suppose
A2:   i < 0;
      then reconsider n1 = -i as Element of NAT by INT_1:3;
A3:   a |^ i = (power(K).(a,|.i.|))" by A2,Def5
      .= (a |^ n1)" by A2,ABSVALUE:def 1;
      a |^ n1 <> 0.K by A1,Lm3;
      hence a |^ i <> 0.K by A3,VECTSP_2:13;
    end;
  end;
