
theorem Th3:
  for F being Function, i being set st i in dom F holds rng proj(F,i) c= F.i &
  (product F <> {} implies rng proj(F,i) = F.i)
proof
  let F be Function, i be set;
  assume
A1: i in dom F;
  thus
A2: rng proj(F,i) c= F.i
  proof
    let x be object;
    assume x in rng proj(F,i);
    then consider f9 being object such that
A3: f9 in dom proj(F,i) and
A4: x = proj(F,i).f9 by FUNCT_1:def 3;
    f9 in product F by A3;
    then consider f being Function such that
A5: f9 = f and
    dom f = dom F and
A6: for x being object st x in dom F holds f.x in F.x by CARD_3:def 5;
    proj(F,i).f = f.i by A3,A5,CARD_3:def 16;
    hence thesis by A1,A4,A5,A6;
  end;
  assume
A7: product F <> {};
  thus rng proj(F,i) c= F.i by A2;
  let x be object;
  set f9 = the Element of product F;
  consider f being Function such that
A8: f9 = f and
A9: dom f = dom F and
  for x being object st x in dom F holds f.x in F.x by A7,CARD_3:def 5;
  assume x in F.i;
  then f+*(i,x)in product F by A7,A8,Th2;
  then
A10: f+*(i,x) in dom proj(F,i) by CARD_3:def 16;
  (f+*(i,x)).i = x by A1,A9,FUNCT_7:31;
  then proj(F,i).(f+*(i,x)) = x by A10,CARD_3:def 16;
  hence thesis by A10,FUNCT_1:def 3;
end;
