let n be Nat; :: thesis: for X being BCI-Algebra_with_Condition(S)
for x, a being Element of X holds (x,a) to_power n = x \ (a |^ n)
let X be BCI-Algebra_with_Condition(S); :: thesis: for x, a being Element of X holds (x,a) to_power n = x \ (a |^ n)
let x, a be Element of X; :: thesis: (x,a) to_power n = x \ (a |^ n)
defpred S₁[ set ] means for m being Nat st m = $1 & m <= n holds
(x,a) to_power m = x \ (a |^ m);

now :: thesis: for k being Nat st ( for m being Nat st m = k & m <= n holds
(x,a) to_power m = x \ (a |^ m) ) holds
for m being Nat st m = k + 1 & m <= n holds
(x,a) to_power m = x \ (a |^ m)

let k be Nat; :: thesis: ( ( for m being Nat st m = k & m <= n holds
(x,a) to_power m = x \ (a |^ m) ) implies for m being Nat st m = k + 1 & m <= n holds
(x,a) to_power m = x \ (a |^ m) )
assume A1: for m being Nat st m = k & m <= n holds
(x,a) to_power m = x \ (a |^ m) ; :: thesis: for m being Nat st m = k + 1 & m <= n holds
(x,a) to_power m = x \ (a |^ m)
let m be Nat; :: thesis: ( m = k + 1 & m <= n implies (x,a) to_power m = x \ (a |^ m) )
assume that
A2: m = k + 1 and
A3: m <= n ; :: thesis: (x,a) to_power m = x \ (a |^ m)
A4: ( (x,a) to_power m = ((x,a) to_power k) \ a & k <= n ) by A2, A3, BCIALG_2:4, NAT_1:13;
x \ (a |^ m) = x \ ((a |^ k) * a) by A2, Def6
.= (x \ (a |^ k)) \ a by Th11 ;
hence (x,a) to_power m = x \ (a |^ m) by A1, A4; :: thesis: verum

end;

then A5: for k being Nat st S₁[k] holds
S₁[k + 1] ;
A6: S₁[ 0 ] by Lm7;
for n being Nat holds S₁[n] from NAT_1:sch 2(A6, A5);
hence (x,a) to_power n = x \ (a |^ n) ; :: thesis: verum