let A, B, C, D be complex-membered set ; :: 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 a be Complex; :: according to MEMBERED:def 7 :: thesis: ( not a in A ** C or a in B ** D )
assume a in A ** C ; :: thesis: a in B ** D
then ex c, c1 being Complex st
( a = c * c1 & c in A & c1 in C ) ;
hence a in B ** D by A1; :: thesis: verum