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 Th2:
  [|M,M|] is Equivalence_Relation of M
proof
  set J = [|M,M|];
  for i be set st i in I holds J.i is Relation of M.i
  proof
    let i be set;
    assume i in I;
    then J.i c= [:M.i,M.i:] by PBOOLE:def 16;
    hence thesis;
  end;
  then reconsider J as ManySortedRelation of M by MSUALG_4:def 1;
  for i be object, R be Relation of M.i st i in I & J.i = R holds R is
  Equivalence_Relation of M.i
  proof
    let i be object;
    let R be Relation of M.i;
    assume that
A1: i in I and
A2: J.i = R;
    J.i = [:M.i,M.i:] by A1,PBOOLE:def 16
      .= nabla M.i by EQREL_1:def 1;
    hence thesis by A2,EQREL_1:4;
  end;
  hence thesis by MSUALG_4:def 2;
end;
