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;
reserve v for Valuation of K;

theorem Th29:
  K is having_valuation & a <> 0.K implies v.(a|^i) = i * v.a
  proof
    assume that
A1: K is having_valuation and
A2: a <> 0.K;
    defpred P[Nat] means v.(a|^$1) = $1 * v.a;
    a|^0 = 1_K by GROUP_1:def 7;
    then
A3: P[0] by A1,Th17;
A4: P[n] implies P[n+1]
    proof
      assume
A5:   P[n];
A6:   v.a in REAL by A1,A2,Th18,XXREAL_0:14;
      reconsider N = n as ExtReal;
      thus v.(a|^(n+1)) = v.(a|^n*a) by Lm2
      .= n * v.a + v.a by A5,A1,Def8
      .= n * v.a + 1. * v.a by XXREAL_3:81
      .= v.a * (N+1.) by A6,XXREAL_3:95
      .= (n+1) * v.a by XXREAL_3:def 2;
    end;
A7: P[n] from NAT_1:sch 2(A3,A4);
    per cases;
    suppose i >= 0;
      then reconsider j = i as Element of NAT by INT_1:3;
      P[j] by A7;
      hence thesis;
    end;
    suppose
A8:   i < 0;
      then reconsider n1 = -i as Element of NAT by INT_1:3;
      reconsider I = i as ExtReal;
A9:   v.(a |^ i) = v.((power(K).(a,|.i.|))") by A8,Def5
      .= v.((a |^ n1)") by A8,ABSVALUE:def 1;
      v.((a |^ n1)") = -v.(a |^ n1) by A1,A2,Th21,Lm3
      .= -n1*v.a by A7
      .= -((-I)*v.a) by XXREAL_3:def 3
      .= --i*v.a by XXREAL_3:92;
      hence thesis by A9;
    end;
  end;
