let X be BCI-algebra; :: thesis: for x being Element of X
for n, m being Nat holds (((0. X),(((0. X),x) to_power m)) to_power n) ` = ((0. X),x) to_power (m * n)
let x be Element of X; :: thesis: for n, m being Nat holds (((0. X),(((0. X),x) to_power m)) to_power n) ` = ((0. X),x) to_power (m * n)
let n, m be Nat; :: thesis: (((0. X),(((0. X),x) to_power m)) to_power n) ` = ((0. X),x) to_power (m * n)
defpred S<sub>1</sub>[ set ] means for j being Nat st j = $1 & j <= n holds
(((0. X),(((0. X),x) to_power m)) to_power j) ` = ((0. X),x) to_power (m * j);

now :: thesis: for k being Nat st ( for j being Nat st j = k & j <= n holds
(((0. X),(((0. X),x) to_power m)) to_power j) ` = ((0. X),x) to_power (m * j) ) holds
for j being Nat st j = k + 1 & j <= n holds
(((0. X),(((0. X),x) to_power m)) to_power (k + 1)) ` = ((0. X),x) to_power (m * (k + 1))

let k be Nat; :: thesis: ( ( for j being Nat st j = k & j <= n holds
(((0. X),(((0. X),x) to_power m)) to_power j) ` = ((0. X),x) to_power (m * j) ) implies for j being Nat st j = k + 1 & j <= n holds
(((0. X),(((0. X),x) to_power m)) to_power (k + 1)) ` = ((0. X),x) to_power (m * (k + 1)) )
assume A1: for j being Nat st j = k & j <= n holds
(((0. X),(((0. X),x) to_power m)) to_power j) ` = ((0. X),x) to_power (m * j) ; :: thesis: for j being Nat st j = k + 1 & j <= n holds
(((0. X),(((0. X),x) to_power m)) to_power (k + 1)) ` = ((0. X),x) to_power (m * (k + 1))
let j be Nat; :: thesis: ( j = k + 1 & j <= n implies (((0. X),(((0. X),x) to_power m)) to_power (k + 1)) ` = ((0. X),x) to_power (m * (k + 1)) )
assume ( j = k + 1 & j <= n ) ; :: thesis: (((0. X),(((0. X),x) to_power m)) to_power (k + 1)) ` = ((0. X),x) to_power (m * (k + 1))
then A2: k <= n by NAT_1:13;
(((0. X),(((0. X),x) to_power m)) to_power (k + 1)) ` = ((((0. X),(((0. X),x) to_power m)) to_power k) \ (((0. X),x) to_power m)) ` by Th4
.= ((((0. X),(((0. X),x) to_power m)) to_power k) `) \ ((((0. X),x) to_power m) `) by BCIALG_1:9
.= (((0. X),x) to_power (m * k)) \ ((((0. X),x) to_power m) `) by A1, A2
.= ((0. X),x) to_power ((m * k) + m) by Th13 ;
hence (((0. X),(((0. X),x) to_power m)) to_power (k + 1)) ` = ((0. X),x) to_power (m * (k + 1)) ; :: thesis: verum

end;

then A3: for k being Nat st S₁[k] holds
S₁[k + 1] ;
(((0. X),(((0. X),x) to_power m)) to_power 0) ` = (0. X) ` by Th1
.= 0. X by BCIALG_1:def 5 ;
then A4: S₁[ 0 ] by Th1;
for n being Nat holds S₁[n] from NAT_1:sch 2(A4, A3);
hence (((0. X),(((0. X),x) to_power m)) to_power n) ` = ((0. X),x) to_power (m * n) ; :: thesis: verum