let R be non empty unital associative multMagma ; :: thesis: for a being Element of R
for n, m being Element of NAT holds (a |^ n) |^ m = a |^ (n * m)
let a be Element of R; :: thesis: for n, m being Element of NAT holds (a |^ n) |^ m = a |^ (n * m)
let n, m be Element of NAT ; :: thesis: (a |^ n) |^ m = a |^ (n * m)
defpred S₁[ Element of NAT ] means (power R) . (a |^ n),$1 = (power R) . a,(n * $1);

A1: now

let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume S₁[m] ; :: thesis: S₁[m + 1]
then (power R) . (a |^ n),(m + 1) = (a |^ (n * m)) * (a |^ n) by GROUP_1:def 8
.= a |^ ((n * m) + n) by Th11
.= (power R) . a,(n * (m + 1)) ;
hence S₁[m + 1] ; :: thesis: verum

end;

(power R) . (a |^ n),0 = 1_ R by GROUP_1:def 8
.= (power R) . a,(n * 0 ) by GROUP_1:def 8 ;
then A2: S₁[ 0 ] ;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A2, A1);
hence (a |^ n) |^ m = a |^ (n * m) ; :: thesis: verum