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

A + c= B +

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

assume A c= B ; :: thesis: A + c= B +

then A |^.. 1 c= B |^.. 1 by Th13;

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

hence A + c= B + by Th50; :: thesis: verum

A + c= B +

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

assume A c= B ; :: thesis: A + c= B +

then A |^.. 1 c= B |^.. 1 by Th13;

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

hence A + c= B + by Th50; :: thesis: verum