reserve i, j, k, c, m, n for Nat,
  a, x, y, z, X, Y for set,
  D, E for non empty set,
  R for Relation,
  f, g for Function,
  p, q for FinSequence;

theorem Th7:
 for f being Function-yielding Function holds
  {} in rng f implies <:f:> = {}
proof let f be Function-yielding Function;
A1: dom <:f:> = meet doms f by FUNCT_6:29
    .= meet rng doms f by FUNCT_6:def 4;
  assume {} in rng f;
  then consider x being object such that
A2: x in dom f and
A3: f.x = {} by FUNCT_1:def 3;
A4: dom doms f = dom f by FUNCT_6:def 2;
  then
A5: x in dom doms f by A2;
  then (doms f).x = {} by A3,A4,FUNCT_6:def 2;
  hence thesis by A1,A5,FUNCT_1:3,SETFAM_1:4;
end;
