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

A1: now

let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A2: S₁[m] ; :: thesis: S₁[m + 1]
(power R) . a,(n + (m + 1)) = (power R) . a,((n + m) + 1)
.= (((power R) . a,n) * ((power R) . a,m)) * a by A2, GROUP_1:def 8
.= ((power R) . a,n) * (((power R) . a,m) * a) by GROUP_1:def 4
.= ((power R) . a,n) * ((power R) . a,(m + 1)) by GROUP_1:def 8 ;
hence S₁[m + 1] ; :: thesis: verum

end;

(power R) . a,(n + 0 ) = ((power R) . a,n) * (1_ R) by GROUP_1:def 5
.= ((power R) . a,n) * ((power R) . a,0 ) by GROUP_1:def 8 ;
then A3: S₁[ 0 ] ;
for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A3, A1);
hence a |^ (n + m) = (a |^ n) * (a |^ m) ; :: thesis: verum