reserve I, G, H for set, i, x for object,
  A, B, M for ManySortedSet of I,
  sf, sg, sh for Subset-Family of I,
  v, w for Subset of I,
  F for ManySortedFunction of I;
reserve X, Y, Z for ManySortedSet of I;
reserve SF, SG, SH for MSSubsetFamily of M,
  SFe for non-empty MSSubsetFamily of M,
  V, W for ManySortedSubset of M;

theorem Th51: :: SETFAM_1:12
  SF = { V,W } implies meet SF = V (/\) W
proof
  assume
A1: SF = { V,W };
  now
    let i be object;
    assume
A2: i in I;
    then
    ex Q be Subset-Family of (M.i) st Q = SF.i & (meet SF).i = Intersect Q
    by Def1;
    hence (meet SF).i = meet ({V,W}.i) by A1,A2,SETFAM_1:def 9
      .= meet {V.i,W.i} by A2,PZFMISC1:def 2
      .= V.i /\ W.i by SETFAM_1:11
      .= (V (/\) W).i by A2,PBOOLE:def 5;
  end;
  hence thesis;
end;
