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 :: SETFAM_1:7 :: SETFAM_1:59
  SF c= SG implies meet SG c= meet SF
proof
  assume
A1: SF c= SG;
  let i be object;
  assume
A2: i in I;
  then consider Qf be Subset-Family of (M.i) such that
A3: Qf = SF.i and
A4: (meet SF).i = Intersect Qf by Def1;
  consider Qg be Subset-Family of (M.i) such that
A5: Qg = SG.i and
A6: (meet SG).i = Intersect Qg by A2,Def1;
  Qf c= Qg by A1,A2,A3,A5;
  hence thesis by A4,A6,SETFAM_1:44;
end;
