
theorem Th37:
  for f being non-empty Function, X being set, i, x being object
  for g being Function st i in dom f & x in X & g in product f
  holds g +* (i,x) in product(f +* (i,X))
proof
  let f be non-empty Function, X be set, i,x be object;
  let g be Function;
  assume A1: i in dom f & x in X & g in product f;
  A2: dom(g +* (i,x)) = dom g by FUNCT_7:30
    .= dom f by A1, CARD_3:9
    .= dom(f +* (i,X)) by FUNCT_7:30;
  for y being object st y in dom(f+*(i,X)) holds (g+*(i,x)).y in (f+*(i,X)).y
  proof
    let y be object;
    assume A3: y in dom(f+*(i,X));
    then A4:y in dom f by FUNCT_7:30;
    per cases;
    suppose A5: i = y;
      y in dom f by A3, FUNCT_7:30;
      then y in dom g by A1, CARD_3:9;
      then (g+*(i,x)).y = x by A5, FUNCT_7:31;
      hence thesis by A1, A5, FUNCT_7:31;
    end;
    suppose i <> y;
      then (g+*(i,x)).y = g.y & (f+*(i,X)).y = f.y by FUNCT_7:32;
      hence thesis by A4,A1,CARD_3:9;
    end;
  end;
  hence thesis by A2, CARD_3:9;
end;
