
theorem Th5:
  for F,f being Function, i,xi being set st xi in F.i & i in dom F
  & f in product F holds f+*(i,xi) in proj(F,i)"({xi})
proof
  let F,f be Function, i,xi be set;
  assume that
A1: xi in F.i and
A2: i in dom F and
A3: f in product F;
  (f+*(i,xi))in product F by A1,A3,Th2;
  then
A4: (f+*(i,xi))in dom proj(F,i) by CARD_3:def 16;
  i in dom f by A2,A3,CARD_3:9;
  then (f+*(i,xi)).i = xi by FUNCT_7:31;
  then (f+*(i,xi)).i in {xi} by TARSKI:def 1;
  then proj(F,i).(f+*(i,xi)) in {xi} by A4,CARD_3:def 16;
  hence thesis by A4,FUNCT_1:def 7;
end;
