defpred S₁[ Nat] means not S ^^ S is empty ;
A1: S₁[ 0 ] ;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
set T = S ^^ k;
assume not S ^^ k is empty ; :: thesis: S₁[k + 1]
then reconsider T = S ^^ k as non empty FinSequence-membered set ;
S ^^ (k + 1) = T ^ S by Th7;
hence S₁[k + 1] ; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A2);
hence ( not S ^^ n is empty & S ^^ n is FinSequence-membered ) ; :: thesis: verum