let n be Nat; :: thesis: for G being Group holds (1_ G) |^ n = 1_ G
let G be Group; :: thesis: (1_ G) |^ n = 1_ G
defpred S₁[ Nat] means (1_ G) |^ $1 = 1_ G;

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 (1_ G) |^ (n + 1) = (1_ G) * (1_ G) by Def7
.= 1_ G by Def4 ;
hence S₁[n + 1] ; :: thesis: verum

end;

A2: S₁[ 0 ] by Def7;
for n being Nat holds S₁[n] from NAT_1:sch 2(A2, A1);
hence (1_ G) |^ n = 1_ G ; :: thesis: verum