
theorem Th3:
  for M being non void SubsetFamilyStr holds M is subset-closed iff
  for A,B being Subset of M st A is independent & B c= A holds B is independent
proof
  let M be non void SubsetFamilyStr;
  thus M is subset-closed implies for A,B being Subset of M st A is
  independent & B c= A holds B is independent by Th1;
  assume
A1: for A,B being Subset of M st A is independent & B c= A holds B is
  independent;
  let x,y be set;
  assume x in the_family_of M;
  then
A2: x is independent Subset of M by Def2;
  assume y c= x;
  then y is independent Subset of M by A1,A2,XBOOLE_1:1;
  hence thesis by Def2;
end;
