theorem Th21:
  f <> {} implies
  for x being object holds
   x in meet f iff for y being object st y in dom f holds x in f.y
proof
  assume
A1: f <> {};
  let x be object;
  thus x in meet f implies for y being object st y in dom f holds x in f.y
  proof
    assume
A2: x in meet f;
    let y be object;
    assume y in dom f;
    then f.y in rng f by FUNCT_1:def 3;
    then meet f c= f.y by SETFAM_1:3;
    hence thesis by A2;
  end;
  assume
A3: for y being object st y in dom f holds x in f.y;
  now
    let z be set;
    assume z in rng f;
    then ex y being object st y in dom f & z = f.y by FUNCT_1:def 3;
    hence x in z by A3;
  end;
  hence thesis by A1,SETFAM_1:def 1;
end;
