let n be Nat; :: thesis: for K being non empty non degenerated right_complementable distributive Abelian add-associative right_zeroed associative Field-like doubleLoopStr
for v being Valuation of K st K is having_valuation holds
for y being Element of (ValuatRing v) holds (power K) . (y,n) = (power (ValuatRing v)) . (y,n)
let K be non empty non degenerated right_complementable distributive Abelian add-associative right_zeroed associative Field-like doubleLoopStr ; :: thesis: for v being Valuation of K st K is having_valuation holds
for y being Element of (ValuatRing v) holds (power K) . (y,n) = (power (ValuatRing v)) . (y,n)
let v be Valuation of K; :: thesis: ( K is having_valuation implies for y being Element of (ValuatRing v) holds (power K) . (y,n) = (power (ValuatRing v)) . (y,n) )
set R = ValuatRing v;
assume A1: K is having_valuation ; :: thesis: for y being Element of (ValuatRing v) holds (power K) . (y,n) = (power (ValuatRing v)) . (y,n)
let y be Element of (ValuatRing v); :: thesis: (power K) . (y,n) = (power (ValuatRing v)) . (y,n)
defpred S₁[ 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 (ValuatRing v)) . (y,0) = 1_ (ValuatRing v) ) by GROUP_1:def 7;
then A2: S₁[ 0 ] by A1, Def12;
A3: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A4: S₁[n] ; :: thesis: S₁[n + 1]
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 S₁[n + 1] by A1, A4, Th55; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A2, A3);
hence (power K) . (y,n) = (power (ValuatRing v)) . (y,n) ; :: thesis: verum