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[ Nat] means (power R) . (a,(n + $1)) = ((power R) . (a,n)) * (a |^ $1);

A1: now :: thesis: for m being Nat st S₁[m] holds
S₁[m + 1]

let m be Nat; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A2: S₁[m] ; :: thesis: S₁[m + 1]
reconsider mm = m as Element of NAT by ORDINAL1:def 12;
(power R) . (a,(n + (m + 1))) = (power R) . (a,((n + m) + 1))
.= (((power R) . (a,n)) * ((power R) . (a,mm))) * a by A2, GROUP_1:def 7
.= ((power R) . (a,n)) * (((power R) . (a,mm)) * a) by GROUP_1:def 3
.= ((power R) . (a,n)) * (a |^ (m + 1)) by GROUP_1:def 7 ;
hence S₁[m + 1] ; :: thesis: verum

end;

(power R) . (a,(n + 0)) = ((power R) . (a,n)) * (1_ R) by GROUP_1:def 4
.= ((power R) . (a,n)) * ((power R) . (a,0)) by GROUP_1:def 7 ;
then A3: S₁[ 0 ] ;
for m being Nat holds S₁[m] from NAT_1:sch 2(A3, A1);
hence a |^ (n + m) = (a |^ n) * (a |^ m) ; :: thesis: verum