let G be Group; :: thesis: for A, B, C, D being Subset of G st A c= B & C c= D holds
A * C c= B * D
let A, B, C, D be Subset of G; :: thesis: ( A c= B & C c= D implies A * C c= B * D )
assume A1: ( A c= B & C c= D ) ; :: thesis: A * C c= B * D
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in A * C or x in B * D )
assume x in A * C ; :: thesis: x in B * D
then ex a, c being Element of G st
( x = a * c & a in A & c in C ) ;
hence x in B * D by A1; :: thesis: verum