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

for k being Nat holds (A *) ^^ (A |^ k) = A |^.. k

let A be Subset of (E ^omega); :: thesis: for k being Nat holds (A *) ^^ (A |^ k) = A |^.. k

let k be Nat; :: thesis: (A *) ^^ (A |^ k) = A |^.. k

defpred S_{1}[ Nat] means (A *) ^^ (A |^ $1) = A |^.. $1;

.= A * by FLANG_1:13

.= A |^.. 0 by Th11 ;

then A3: S_{1}[ 0 ]
;

for k being Nat holds S_{1}[k]
from NAT_1:sch 2(A3, A1);

hence (A *) ^^ (A |^ k) = A |^.. k ; :: thesis: verum

for k being Nat holds (A *) ^^ (A |^ k) = A |^.. k

let A be Subset of (E ^omega); :: thesis: for k being Nat holds (A *) ^^ (A |^ k) = A |^.. k

let k be Nat; :: thesis: (A *) ^^ (A |^ k) = A |^.. k

defpred S

A1: now :: thesis: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]

(A *) ^^ (A |^ 0) =
(A *) ^^ {(<%> E)}
by FLANG_1:24
S

let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume A2: S_{1}[k]
; :: thesis: S_{1}[k + 1]

(A *) ^^ (A |^ (k + 1)) = (A *) ^^ ((A |^ k) ^^ A) by FLANG_1:23

.= (A |^.. k) ^^ A by A2, FLANG_1:18

.= A |^.. (k + 1) by Th16 ;

hence S_{1}[k + 1]
; :: thesis: verum

end;assume A2: S

(A *) ^^ (A |^ (k + 1)) = (A *) ^^ ((A |^ k) ^^ A) by FLANG_1:23

.= (A |^.. k) ^^ A by A2, FLANG_1:18

.= A |^.. (k + 1) by Th16 ;

hence S

.= A * by FLANG_1:13

.= A |^.. 0 by Th11 ;

then A3: S

for k being Nat holds S

hence (A *) ^^ (A |^ k) = A |^.. k ; :: thesis: verum