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_1:9
  A in SF & A (/\) B = EmptyMS I implies meet SF (/\) B = EmptyMS I
proof
  assume that
A1: A in SF and
A2: A (/\) B = EmptyMS I;
  now
    let i be object;
    assume
A3: i in I;
    then consider Q be Subset-Family of (M.i) such that
A4: Q = SF.i and
A5: (meet SF).i = Intersect Q by Def1;
A6: A.i in SF.i by A1,A3;
    A.i /\ B.i = EmptyMS I.i by A2,A3,PBOOLE:def 5;
    then A.i /\ B.i = {};
    then A.i misses B.i;
    then meet Q misses B.i by A4,A6,SETFAM_1:8;
    then meet Q /\ B.i = {};
    then
A7: meet Q /\ B.i = EmptyMS I.i;
    Intersect Q = meet Q by A4,A6,SETFAM_1:def 9;
    hence (meet SF (/\) B).i = EmptyMS I.i by A3,A5,A7,PBOOLE:def 5;
  end;
  hence thesis;
end;
