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 z be Complex; :: according to MEMBERED:def 7 :: thesis: ( not z in A /// C or z in B /// D )
A2: A /// C = { (c1 / c2) where c1, c2 is Complex : ( c1 in A & c2 in C ) } by Th115;
assume z in A /// C ; :: thesis: z in B /// D
then ex c, c1 being Complex st
( z = c / c1 & c in A & c1 in C ) by A2;
hence z in B /// D by A1, Th116; :: thesis: verum