reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem
  not {} in rng f implies Funcs(f,{}) = dom f --> {}
proof
  assume
A1: not {} in rng f;
A2: now
    let x be object;
    assume
 x in dom f;
    then
A4: f.x <> {} by A1,FUNCT_1:def 3;
    thus (dom f --> {}).x = {}
      .= Funcs(f.x,{}) by A4;
  end;
  dom (dom f --> {}) = dom f;
  hence thesis by A2,Def7;
end;
