let P, Q be FinSequence-membered set ; :: thesis: for n being Nat st Q c= P * holds
Q ^^ n c= P *
let n be Nat; :: thesis: ( Q c= P * implies Q ^^ n c= P * )
assume A1: Q c= P * ; :: thesis: Q ^^ n c= P *
defpred S₁[ Nat] means Q ^^ $1 c= P * ;
A2: S₁[ 0 ]

proof

Q ^^ 0 = {{}} by Th6
.= P ^^ 0 by Th6 ;
hence S₁[ 0 ] by Th8; :: thesis: verum

end;

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 (Q ^^ k) ^ Q c= P * by A1, Th13;
hence S₁[k + 1] by Th6; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A2, A3);
hence Q ^^ n c= P * ; :: thesis: verum