let X be set ; :: thesis: for A, B being Subset-Family of X st A c= B holds
UniCl A c= UniCl B
let A, B be Subset-Family of X; :: thesis: ( A c= B implies UniCl A c= UniCl B )
assume A1: A c= B ; :: thesis: UniCl A c= UniCl B
let x be object ; :: 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 x9 = x as Subset of X ;
consider T being Subset-Family of X such that
A3: T c= A and
A4: x9 = union T by A2, Def1;
T c= B by A1, A3;
hence x in UniCl B by A4, Def1; :: thesis: verum