let X be BCI-algebra; :: thesis: for x being Element of X
for n being Nat holds (x `) |^ n = (x |^ n) `
let x be Element of X; :: thesis: for n being Nat holds (x `) |^ n = (x |^ n) `
let n be Nat; :: thesis: (x `) |^ n = (x |^ n) `
defpred S₁[ Nat] means (x `) |^ $1 = (x |^ $1) ` ;

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

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume S₁[n] ; :: thesis: S₁[n + 1]
then (x `) |^ (n + 1) = (x `) \ (((x |^ n) `) `) by Th2
.= (x \ ((x |^ n) `)) ` by BCIALG_1:9
.= (x |^ (n + 1)) ` by Th2 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(x `) |^ 0 = 0. X by Def1
.= (0. X) ` by BCIALG_1:def 5 ;
then A2: S₁[ 0 ] by Def1;
for n being Nat holds S₁[n] from NAT_1:sch 2(A2, A1);
hence (x `) |^ n = (x |^ n) ` ; :: thesis: verum