reserve i,j,e,u for object;
reserve I for set; 
reserve x,X,Y,Z,V for ManySortedSet of I;
reserve I for non empty set,
  x,X,Y for ManySortedSet of I;
reserve I for set,
  x,X,Y,Z for ManySortedSet of I;
reserve X for non-empty ManySortedSet of I;

theorem
  for M being ManySortedSet of I, i st i in I holds M.i is Component of M
proof
  let M be ManySortedSet of I, i;
  assume
A1: i in I;
  dom M = I by PARTFUN1:def 2;
  hence thesis by A1,FUNCT_1:def 3;
end;
