
theorem Th14:
  for X being set, A,B being Subset-Family of X holds A c= B
  implies FinMeetCl A c= FinMeetCl B
proof
  let X be set, A,B be Subset-Family of X such that
A1: A c= B;
  let x be object;
  assume
A2: x in FinMeetCl A;
  then reconsider x as Subset of X;
  consider y being Subset-Family of X such that
A3: y c= A and
A4: y is finite & x = Intersect y by A2,Def3;
  y c= B by A1,A3;
  hence thesis by A4,Def3;
end;
