defpred S₁[ Nat] means B ^^ B is antichain-like ;
A2: S₁[ 0 ] ;
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
then (B ^^ k) ^ B is antichain-like ;
hence S₁[k + 1] by Th6; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A2, A3);
hence B ^^ n is antichain-like ; :: thesis: verum