let s be Nat; :: thesis: 1 |^ s = 1
defpred S₁[ Nat] means 1 |^ $1 = 1;

A1: now

let s be Nat; :: thesis: ( S₁[s] implies S₁[s + 1] )
assume S₁[s] ; :: thesis: S₁[s + 1]
then 1 |^ (s + 1) = 1 * 1 by Th11;
hence S₁[s + 1] ; :: thesis: verum

end;

A2: S₁[ 0 ] by RVSUM_1:124;
for s being Nat holds S₁[s] from NAT_1:sch 2(A2, A1);
hence 1 |^ s = 1 ; :: thesis: verum