let L be non empty well-unital doubleLoopStr ; :: thesis: for n being Element of NAT holds (1. L) |^ n = 1. L
let n be Element of NAT ; :: thesis: (1. L) |^ n = 1. L
defpred S₁[ Nat] means (1. L) |^ $1 = 1_ L;

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

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: S₁[k] ; :: thesis: S₁[k + 1]
(1. L) |^ (k + 1) = ((1. L) |^ k) * (1. L) by GROUP_1:def 7
.= 1. L by A2 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A3: S₁[ 0 ] by BINOM:8;
for k being Nat holds S₁[k] from NAT_1:sch 2(A3, A1);
hence (1. L) |^ n = 1. L ; :: thesis: verum