let X be set ; :: thesis: for A, B being Subset-Family of st A c= B holds
UniCl A c= UniCl B
let A, B be Subset-Family of ; :: thesis: ( A c= B implies UniCl A c= UniCl B )
assume A1: A c= B ; :: thesis: UniCl A c= UniCl B
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in UniCl A or x in UniCl B )
assume A2: x in UniCl A ; :: thesis: x in UniCl B
then reconsider x' = x as Subset of ;
consider T being Subset-Family of such that
A3: T c= A and
A4: x' = union T by A2, Def1;
T c= B by A1, A3, XBOOLE_1:1;
hence x in UniCl B by A4, Def1; :: thesis: verum