
theorem Th4:
  for F being Function, i being set st i in dom F holds
  proj(F,i)"(F.i) = product F
proof
  let F be Function, i be set;
  assume
A1: i in dom F;
  dom proj(F,i)=product F by CARD_3:def 16;
  hence proj(F,i)"(F.i) c= product F by RELAT_1:132;
  let f9 be object;
  assume
A2: f9 in product F;
  then consider f being Function such that
A3: f9 = f and
  dom f = dom F and
A4: for x being object st x in dom F holds f.x in F.x by CARD_3:def 5;
A5: f in dom proj(F,i) by A2,A3,CARD_3:def 16;
  f.i in F.i by A1,A4;
  then proj(F,i).f in F.i by A5,CARD_3:def 16;
  hence thesis by A3,A5,FUNCT_1:def 7;
end;
