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]
set m = - (n + 1);
x |^ (- (n + 1)) = (BCI-power X) . ((x `),|.(- (n + 1)).|) by Def2
.= (BCI-power X) . ((((x `) `) `),|.(- (n + 1)).|) by BCIALG_1:8
.= ((x `) `) |^ (- (n + 1)) by Def2 ;
hence S₁[n + 1] ; :: thesis: verum

end;

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