
theorem
  for A being non empty set for L being non empty Sublattice of
EqRelLATT A holds L is trivial or ex e being Equivalence_Relation of A st e in
  the carrier of L & e <> id A
proof
  let A be non empty set;
  let L be non empty Sublattice of EqRelLATT A;
  now
    assume
A1: not ex e being Equivalence_Relation of A st e in the carrier of L
    & e <> id A;
    thus L is trivial
    proof
      consider x be object such that
A2:   x in the carrier of L by XBOOLE_0:def 1;
      the carrier of L c= the carrier of EqRelLATT A by YELLOW_0:def 13;
      then reconsider e=x as Equivalence_Relation of A by A2,LATTICE5:4;
      the carrier of L = {x}
      proof
        thus the carrier of L c= {x}
        proof
          let a be object;
          assume
A3:       a in the carrier of L;
          the carrier of L c= the carrier of EqRelLATT A by YELLOW_0:def 13;
          then reconsider B=a as Equivalence_Relation of A by A3,LATTICE5:4;
          B = id A by A1,A3
            .= e by A1,A2;
          hence thesis by TARSKI:def 1;
        end;
        let A be object;
        assume A in {x};
        hence thesis by A2,TARSKI:def 1;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
