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 Th6:
  for a,b be Element of EqRelLatt M, A,B be Equivalence_Relation of
  M st a = A & b = B holds a [= b iff A c= B
proof
  let a,b be Element of EqRelLatt M;
  let A,B be Equivalence_Relation of M;
  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 M).(A,B) by MSUALG_5:def 5
      .= a "/\" b by A1,A2,LATTICES:def 2
      .= A by A1,A3,LATTICES:4;
    hence thesis by PBOOLE:15;
  end;
  thus A c= B implies a [= b
  proof
    assume
A4: A c= B;
    a "/\" b = (the L_meet of EqRelLatt M).(A,B) by A1,A2,LATTICES:def 2
      .= A (/\) B by MSUALG_5:def 5
      .= a by A1,A4,PBOOLE:23;
    hence thesis by LATTICES:4;
  end;
end;
