reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve Y,x,y,i for set;
reserve r,r1,r2 for Real;

theorem Th7:
  for X be Subset of EqRelLatt M, X1 be SubsetFamily of [|M,M|] st
X1 = X for a,b be Equivalence_Relation of M st a = meet |:X1:| & b in X holds a
  c= b
proof
  let X be Subset of EqRelLatt M;
  let X1 be SubsetFamily of [|M,M|] such that
A1: X1 = X;
  let a,b be Equivalence_Relation of M such that
A2: a = meet |:X1:| and
A3: b in X;
  now
    reconsider b9 = b as Element of Bool [|M,M|] by A1,A3;
    let i be object;
    assume
A4: i in I;
    then
    |:X1:|.i = { x.i where x is Element of Bool [|M,M|] : x in X1 } by A1,A3,
CLOSURE2:14;
    then
A5: b9.i in |:X1:|.i by A1,A3;
    then
A6: for y being object st y in meet (|:X1:|.i) holds y in b.i
         by SETFAM_1:def 1;
    ex Q be Subset-Family of ([|M,M|].i) st Q = |:X1:|.i & meet |:X1:|.i =
    Intersect Q by A4,MSSUBFAM:def 1;
    then a.i = meet (|:X1:|.i) by A2,A5,SETFAM_1:def 9;
    hence a.i c= b.i by A6;
  end;
  hence thesis by PBOOLE:def 2;
end;
