defpred S₁[ Nat] means (1. R) |^ $1 = 1. R;
(1. R) |^ 0 = 1_ R by BINOM:8;
then IA: S₁[ 0 ] ;

IS: 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 IV: S₁[k] ; :: thesis: S₁[k + 1]
(1. R) |^ (k + 1) = ((1. R) |^ k) * ((1. R) |^ 1) by BINOM:10
.= (1. R) * (1. R) by IV, BINOM:8 ;
hence S₁[k + 1] ; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(IA, IS);
hence (1. R) |^ k = 1. R ; :: thesis: verum