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_1:10
  SH = SF (\/) SG implies meet SH = meet SF (/\) meet SG
proof
  assume
A1: SH = SF (\/) SG;
  now
    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 Qh be Subset-Family of (M.i) such that
A5: Qh = SH.i and
A6: (meet SH).i = Intersect Qh by A2,Def1;
    consider Qg be Subset-Family of (M.i) such that
A7: Qg = SG.i and
A8: (meet SG).i = Intersect Qg by A2,Def1;
A9: Qh = Qf \/ Qg by A1,A2,A3,A7,A5,PBOOLE:def 4;
    now
      per cases;
      case
A10:    Qf <> {} & Qg <> {};
        hence (meet SH).i = meet Qh by A6,A9,SETFAM_1:def 9
          .= meet Qf /\ meet Qg by A9,A10,SETFAM_1:9
          .= (meet SF).i /\ meet Qg by A4,A10,SETFAM_1:def 9
          .= (meet SF).i /\ (meet SG).i by A8,A10,SETFAM_1:def 9
          .= (meet SF (/\) meet SG).i by A2,PBOOLE:def 5;
      end;
      case
A11:    Qf <> {} & Qg = {};
        hence (meet SH).i = (meet SF).i /\ M.i by A4,A6,A9,XBOOLE_1:28
          .= (meet SF).i /\ (meet SG).i by A8,A11,SETFAM_1:def 9
          .= (meet SF (/\) meet SG).i by A2,PBOOLE:def 5;
      end;
      case
A12:    Qf = {} & Qg <> {};
        hence (meet SH).i = M.i /\ (meet SG).i by A8,A6,A9,XBOOLE_1:28
          .= (meet SF).i /\ (meet SG).i by A4,A12,SETFAM_1:def 9
          .= (meet SF (/\) meet SG).i by A2,PBOOLE:def 5;
      end;
      case
        Qf = {} & Qg = {};
        hence (meet SH).i = Intersect Qf /\ Intersect Qg by A6,A9
          .= (meet SF (/\) meet SG).i by A2,A4,A8,PBOOLE:def 5;
      end;
    end;
    hence (meet SH).i = (meet SF (/\) meet SG).i;
  end;
  hence thesis;
end;
