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

theorem
  for S be cap-finite-partition-closed Subset-Family of X holds
  for SM be finite Subset of S ex P be finite Subset of S st
  P is a_partition of meet SM
  proof
    let S be cap-finite-partition-closed Subset-Family of X;
    let SM be finite Subset of S;
    consider SF be FinSequence such that
A1: rng SF = SM by FINSEQ_1:52;
    SF is FinSequence of S by A1,FINSEQ_1:def 4;
    hence thesis by A1,Lem8;
  end;
