
theorem Th1:
  for M being non void subset-closed SubsetFamilyStr for A being
independent Subset of M for B being set st B c= A holds B is independent Subset
  of M
proof
  let M be non void subset-closed SubsetFamilyStr;
  let A be independent Subset of M;
  let B be set;
  assume
A1: B c= A;
  A in the_family_of M by Def2;
  then B in the_family_of M by A1,CLASSES1:def 1;
  hence thesis by Def2;
end;
