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 Th60:
  K is having_valuation implies
  for y being Element of ValuatRing v holds
  power(K).(y,n) = power(ValuatRing v).(y,n)
  proof
    set R = ValuatRing v;
    assume
A1: K is having_valuation;
    let y be Element of R;
    defpred P[Nat] means
    power(K).(y,$1) = power(ValuatRing v).(y,$1);
    reconsider x = y as Element of K by A1,Th51;
    power(K).(x,0) = 1_K & power(R).(y,0) = 1_R by GROUP_1:def 7; then
A2: P[0] by A1,Def12;
A3: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume
A4:   P[n];
      reconsider m = n as Element of NAT by ORDINAL1:def 12;
      power(K).(y,n+1) = (power(K).(x,m))*x &
      power(ValuatRing v).(y,n+1) = (power(ValuatRing v)).(y,m)*y
      by GROUP_1:def 7;
      hence P[n+1] by A1,A4,Th55;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
