let E, x be set ; :: thesis: for A being Subset of (E ^omega) st ( x in (A *) ^^ A or x in A ^^ (A *) ) holds
x in A *
let A be Subset of (E ^omega); :: thesis: ( ( x in (A *) ^^ A or x in A ^^ (A *) ) implies x in A * )

A1: now :: thesis: for x being set st x in A ^^ (A *) holds
x in A *

let x be set ; :: thesis: ( x in A ^^ (A *) implies x in A * )
assume x in A ^^ (A *) ; :: thesis: x in A *
then consider a, b being Element of E ^omega such that
A2: a in A and
A3: b in A * and
A4: x = a ^ b by Def1;
consider n being Nat such that
A5: b in A |^ n by A3, Th41;
a in A |^ 1 by A2, Th25;
then a ^ b in A |^ (n + 1) by A5, Th40;
hence x in A * by A4, Th41; :: thesis: verum

end;

now :: thesis: for x being set st x in (A *) ^^ A holds
x in A *

let x be set ; :: thesis: ( x in (A *) ^^ A implies x in A * )
assume x in (A *) ^^ A ; :: thesis: x in A *
then consider a, b being Element of E ^omega such that
A6: a in A * and
A7: b in A and
A8: x = a ^ b by Def1;
consider n being Nat such that
A9: a in A |^ n by A6, Th41;
b in A |^ 1 by A7, Th25;
then a ^ b in A |^ (n + 1) by A9, Th40;
hence x in A * by A8, Th41; :: thesis: verum

end;

hence ( ( x in (A *) ^^ A or x in A ^^ (A *) ) implies x in A * ) by A1; :: thesis: verum