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
  not ex M being non-empty ManySortedSet of I st {} in rng M
proof
  let M be non-empty ManySortedSet of I;
A1: dom M = I by PARTFUN1:def 2;
  assume {} in rng M;
  then ex i being object st i in I & M.i = {} by A1,FUNCT_1:def 3;
  hence contradiction by Def13;
end;
