
theorem Th9:
  for X being set for A, B being Subset-Family of X holds A c= B
  implies UniCl A c= UniCl B
proof
  let X be set;
  let A, B be Subset-Family of X such that
A1: A c= B;
  let x be object;
  assume
A2: x in UniCl A;
  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 thesis by A4,Def1;
end;
