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

theorem Thm3:
  for S be cap-finite-partition-closed diff-c=-finite-partition-closed
  Subset-Family of X, SM,T be finite Subset of S holds
  ex P be finite Subset of S st P is a_partition of (meet SM) \ union T
  proof
    let S be cap-finite-partition-closed diff-c=-finite-partition-closed
    Subset-Family of X;
    let SM,T be finite Subset of S;
    consider RSM be FinSequence such that
A:  SM=rng RSM by FINSEQ_1:52;
    consider RT be FinSequence such that
B:  T=rng RT by FINSEQ_1:52;
C:  RSM is FinSequence of S by A,FINSEQ_1:def 4;
    RT is FinSequence of S by B,FINSEQ_1:def 4;
    then consider P be finite Subset of S such that
D:  P is a_partition of (meet rng RSM) \ Union RT
    by C,Lem9;
    thus thesis by A,B,D;
end;
