
theorem
for X be set, S be non empty Subset-Family of X st
  S is semi-diff-closed holds S is diff-c=-finite-partition-closed
proof
   let X be set, S be non empty Subset-Family of X;
   assume S is semi-diff-closed; then
   for S1,S2 be Element of S st S2 c= S1 holds
    ex x be finite Subset of S st x is a_partition of S1 \ S2 by Lm1;
   hence S is diff-c=-finite-partition-closed by SRINGS_1:def 3;
end;
