reserve X for set;
reserve S for Subset-Family of X;

theorem Thm2:
  for S be Subset-Family of X st
  S is diff-finite-partition-closed holds
  S is diff-c=-finite-partition-closed
  proof
    let S be Subset-Family of X;
    assume
A1: S is diff-finite-partition-closed;
    let S1,S2 be Element of S;
    assume S2 c= S1;
    per cases;
    suppose
A2:   S1\S2 is empty;
      {} is finite Subset of S & {} is a_partition of {}
      by SUBSET_1:1,EQREL_1:45;
      hence thesis by A2;
    end;
    suppose S1\S2 is non empty;
      then consider x be finite Subset of S such that
A3:   x is a_partition of S1\S2 by A1;
      thus thesis by A3;
    end;
  end;
