let X be BCI-algebra; :: thesis: for x being Element of X
for m, n being Element of 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 m, n being Element of NAT holds ((0. X),((0. X),x to_power m) to_power n) ` = (0. X),x to_power (m * n)
let m, n be Element of 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 Element of 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

let k be Element of NAT ; :: thesis: ( ( for j being Element of 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 Element of 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 Element of 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 Element of 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 Element of 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 Element of 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 Element of NAT holds S₁[n] from NAT_1:sch 1(A4, A3);
hence ((0. X),((0. X),x to_power m) to_power n) ` = (0. X),x to_power (m * n) ; :: thesis: verum