reserve i, x, I for set,
  A, B, M for ManySortedSet of I,
  f, f1 for Function;
reserve SF, SG for SubsetFamily of M;
reserve E, T for Element of Bool M;

theorem Th23: :: SETFAM_1:12
  SF = {E,T} implies meet |:SF:| = E (/\) T
proof
  assume
A1: SF = { E,T };
  now
    reconsider sf1 = SF as non empty SubsetFamily of M by A1;
    let i be object such that
A2: i in I;
    ex Q be Subset-Family of (M.i) st Q = |:sf1:|.i & (meet |:sf1:|).i =
    Intersect Q by A2,MSSUBFAM:def 1;
    hence (meet |:SF:|).i = meet (|:sf1:|.i) by A2,SETFAM_1:def 9
      .= meet {E.i,T.i} by A1,A2,Th20
      .= E.i /\ T.i by SETFAM_1:11
      .= (E (/\) T).i by A2,PBOOLE:def 5;
  end;
  hence thesis;
end;
