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 I being non empty set for M being ManySortedSet of I, A being
  Component of M ex i being object st i in I & A = M.i
proof
  let I be non empty set;
  let M be ManySortedSet of I, A be Component of M;
A1: dom M = I by PARTFUN1:def 2;
  then rng M <> {} by RELAT_1:42;
  hence thesis by A1,FUNCT_1:def 3;
end;
