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 Th8:
  for X be Subset of EqRelLatt M, X1 be SubsetFamily of [|M,M|] st
  X1 = X & X is non empty holds meet |:X1:| is Equivalence_Relation of M
proof
  let X be Subset of EqRelLatt M;
  let X1 be SubsetFamily of [|M,M|];
  set a = meet |:X1:|;
  now
    let i be set;
    assume
A1: i in I;
    a c= [|M,M|] by PBOOLE:def 18;
    then a.i c= [|M,M|].i by A1,PBOOLE:def 2;
    hence a.i is Relation of M.i by A1,PBOOLE:def 16;
  end;
  then reconsider a as ManySortedRelation of M by MSUALG_4:def 1;
  assume that
A2: X1 = X and
A3: X is non empty;
  now
    reconsider X19 = X1 as non empty SubsetFamily of [|M,M|] by A2,A3;
    let i be object, R be Relation of M.i;
    assume that
A4: i in I and
A5: a.i = R;
    reconsider i9 = i as Element of I by A4;
    reconsider b = |:X1:|.i9 as Subset-Family of [:M.i,M.i:] by PBOOLE:def 16;
    consider Q be Subset-Family of ([|M,M|].i) such that
A6: Q = |:X1:|.i and
A7: R = Intersect Q by A4,A5,MSSUBFAM:def 1;
    reconsider Q as Subset-Family of [:M.i,M.i:] by A4,PBOOLE:def 16;
    |:X19:| is non-empty;
    then
A8: Q <> {} by A4,A6,PBOOLE:def 13;
A9: |:X19:|.i = { x.i where x is Element of Bool [|M,M|] : x in X1 } by A4,
CLOSURE2:14;
    now
      let Y;
      assume Y in b;
      then consider z be Element of Bool [|M,M|] such that
A10:  Y = z.i and
A11:  z in X by A2,A9;
      z c= [|M,M|] by PBOOLE:def 18;
      then z.i c= [|M,M|].i by A4,PBOOLE:def 2;
      then reconsider Y1 = Y as Relation of M.i by A4,A10,PBOOLE:def 16;
      z is Equivalence_Relation of M by A11,MSUALG_5:def 5;
      then Y1 is Equivalence_Relation of M.i by A4,A10,MSUALG_4:def 2;
      hence Y is Equivalence_Relation of M.i;
    end;
    then meet b is Equivalence_Relation of M.i by A6,A8,EQREL_1:11;
    hence R is Equivalence_Relation of M.i by A6,A7,A8,SETFAM_1:def 9;
  end;
  hence thesis by MSUALG_4:def 2;
end;
