let X be BCI-algebra; :: thesis: for x, y, z being Element of X
for n being Nat st x <= y holds
(x,z) to_power n <= (y,z) to_power n
let x, y, z be Element of X; :: thesis: for n being Nat st x <= y holds
(x,z) to_power n <= (y,z) to_power n
let n be Nat; :: thesis: ( x <= y implies (x,z) to_power n <= (y,z) to_power n )
defpred S₁[ set ] means for m being Nat st m = $1 & m <= n holds
(x,z) to_power m <= (y,z) to_power m;
A1: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: for m being Nat st m = k & m <= n holds
(x,z) to_power m <= (y,z) to_power m ; :: thesis: S₁[k + 1]
let m be Nat; :: thesis: ( m = k + 1 & m <= n implies (x,z) to_power m <= (y,z) to_power m )
assume that
A3: m = k + 1 and
A4: m <= n ; :: thesis: (x,z) to_power m <= (y,z) to_power m
k <= n by A3, A4, NAT_1:13;
then (x,z) to_power k <= (y,z) to_power k by A2;
then ((x,z) to_power k) \ z <= ((y,z) to_power k) \ z by BCIALG_1:5;
then (x,z) to_power (k + 1) <= ((y,z) to_power k) \ z by Th4;
hence (x,z) to_power m <= (y,z) to_power m by A3, Th4; :: thesis: verum

end;

assume x <= y ; :: thesis: (x,z) to_power n <= (y,z) to_power n
then (x,z) to_power 0 <= y by Th1;
then A5: S₁[ 0 ] by Th1;
for n being Nat holds S₁[n] from NAT_1:sch 2(A5, A1);
hence (x,z) to_power n <= (y,z) to_power n ; :: thesis: verum