let E be set ; :: thesis: for A, B being Subset of (E ^omega) st A c= B + holds

B + = (B \/ A) +

let A, B be Subset of (E ^omega); :: thesis: ( A c= B + implies B + = (B \/ A) + )

assume A c= B + ; :: thesis: B + = (B \/ A) +

then A c= B |^.. 1 by Th50;

then B |^.. 1 = (B \/ A) |^.. 1 by Th47;

then B |^.. 1 = (B \/ A) + by Th50;

hence B + = (B \/ A) + by Th50; :: thesis: verum

B + = (B \/ A) +

let A, B be Subset of (E ^omega); :: thesis: ( A c= B + implies B + = (B \/ A) + )

assume A c= B + ; :: thesis: B + = (B \/ A) +

then A c= B |^.. 1 by Th50;

then B |^.. 1 = (B \/ A) |^.. 1 by Th47;

then B |^.. 1 = (B \/ A) + by Th50;

hence B + = (B \/ A) + by Th50; :: thesis: verum