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

for n being Nat holds A ^^ (A |^.. n) = (A |^.. n) ^^ A

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

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

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

.= (A *) ^^ A by FLANG_1:57

.= (A |^.. 0) ^^ A 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 |^.. n) = (A |^.. n) ^^ A ; :: thesis: verum

for n being Nat holds A ^^ (A |^.. n) = (A |^.. n) ^^ A

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

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

defpred S

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

S_{1}[k + 1]

A ^^ (A |^.. 0) =
A ^^ (A *)
by Th11
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 Th16

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

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

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

end;assume A2: S

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

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

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

hence S

.= (A *) ^^ A by FLANG_1:57

.= (A |^.. 0) ^^ A by Th11 ;

then A3: S

for k being Nat holds S

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