let X be BCI-algebra; :: thesis: for n being Nat holds (0. X) |^ n = 0. X
let n be Nat; :: thesis: (0. X) |^ n = 0. X
defpred S₁[ Nat] means (0. X) |^ $1 = 0. X;

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 (0. X) |^ (n + 1) = ((0. X) `) ` by Th2
.= (0. X) ` by BCIALG_1:def 5
.= 0. X by BCIALG_1:def 5 ;
hence S₁[n + 1] ; :: thesis: verum

end;

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