
theorem Th2:
  for F,f being Function, i,xi being set st xi in F.i & f in product F
  holds f+*(i,xi) in product F
proof
  let F,f be Function, i,xi be set;
  assume
A1: xi in F.i;
  assume
A2: f in product F;
A3: for x being object st x in dom F holds (f+*(i,xi)).x in F.x
  proof
    let x be object;
    assume
A4: x in dom F;
    per cases;
    suppose
A5:   i=x;
      thus (f+*(i,xi)).x in F.x
      proof
        per cases;
        suppose
          i in dom f;
          hence thesis by A1,A5,FUNCT_7:31;
        end;
        suppose
          not i in dom f;
          then (f+*(i,xi)) =f by FUNCT_7:def 3;
          hence thesis by A2,A4,CARD_3:9;
        end;
      end;
    end;
    suppose
      i<>x;
      then (f+*(i,xi)).x = f.x by FUNCT_7:32;
      hence thesis by A2,A4,CARD_3:9;
    end;
  end;
  dom f = dom F by A2,CARD_3:9;
  then dom(f+*(i,xi)) = dom F by FUNCT_7:30;
  hence thesis by A3,CARD_3:9;
end;
