let E be set ; :: thesis: for A, B being Subset of (E ^omega)
for n being Nat st A c= B holds
A |^ n c= B |^ n
let A, B be Subset of (E ^omega); :: thesis: for n being Nat st A c= B holds
A |^ n c= B |^ n
let n be Nat; :: thesis: ( A c= B implies A |^ n c= B |^ n )
defpred S₁[ Nat] means A |^ $1 c= B |^ $1;
assume A1: A c= B ; :: thesis: A |^ n c= B |^ n

A2: now :: thesis: for n being Nat st S₁[n] holds
S₁[n + 1]

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A3: S₁[n] ; :: thesis: S₁[n + 1]
( (A |^ n) ^^ A = A |^ (n + 1) & (B |^ n) ^^ B = B |^ (n + 1) ) by Th23;
hence S₁[n + 1] by A1, A3, Th17; :: thesis: verum

end;

A |^ 0 = {(<%> E)} by Th24;
then A4: S₁[ 0 ] by Th24;
for n being Nat holds S₁[n] from NAT_1:sch 2(A4, A2);
hence A |^ n c= B |^ n ; :: thesis: verum