reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve Y,x,y,y1,i,j for set;
reserve k for Element of NAT;
reserve p for FinSequence;
reserve S for non void non empty ManySortedSign;
reserve A for non-empty MSAlgebra over S;

theorem Th2:
  for a,b be Element of EqRelLatt Y for A,B be Equivalence_Relation
  of Y st a = A & b = B holds a [= b iff A c= B
proof
  let a,b be Element of EqRelLatt Y;
  let A,B be Equivalence_Relation of Y;
  assume that
A1: a = A and
A2: b = B;
  thus a [= b implies A c= B
  proof
    assume
A3: a [= b;
    A /\ B = (the L_meet of EqRelLatt Y).(A,B) by MSUALG_5:def 2
      .= a "/\" b by A1,A2,LATTICES:def 2
      .= A by A1,A3,LATTICES:4;
    hence thesis by XBOOLE_1:17;
  end;
  thus A c= B implies a [= b
  proof
    assume
A4: A c= B;
    a "/\" b = (the L_meet of EqRelLatt Y).(A,B) by A1,A2,LATTICES:def 2
      .= A /\ B by MSUALG_5:def 2
      .= a by A1,A4,XBOOLE_1:28;
    hence thesis by LATTICES:4;
  end;
end;
