let E be set ; :: thesis: for A being Subset of (E ^omega)

for k, l being Nat st k <= l holds

(A *) ^^ (A |^ (k,l)) = A |^.. k

let A be Subset of (E ^omega); :: thesis: for k, l being Nat st k <= l holds

(A *) ^^ (A |^ (k,l)) = A |^.. k

let k, l be Nat; :: thesis: ( k <= l implies (A *) ^^ (A |^ (k,l)) = A |^.. k )

assume k <= l ; :: thesis: (A *) ^^ (A |^ (k,l)) = A |^.. k

then (A |^.. 0) ^^ (A |^ (k,l)) = A |^.. (0 + k) by Th33;

hence (A *) ^^ (A |^ (k,l)) = A |^.. k by Th11; :: thesis: verum

for k, l being Nat st k <= l holds

(A *) ^^ (A |^ (k,l)) = A |^.. k

let A be Subset of (E ^omega); :: thesis: for k, l being Nat st k <= l holds

(A *) ^^ (A |^ (k,l)) = A |^.. k

let k, l be Nat; :: thesis: ( k <= l implies (A *) ^^ (A |^ (k,l)) = A |^.. k )

assume k <= l ; :: thesis: (A *) ^^ (A |^ (k,l)) = A |^.. k

then (A |^.. 0) ^^ (A |^ (k,l)) = A |^.. (0 + k) by Th33;

hence (A *) ^^ (A |^ (k,l)) = A |^.. k by Th11; :: thesis: verum