
theorem Th21:
  for f, g being non-empty Function, i, x being object
  st x in product f /\ product(f+*g) holds proj(f,i).x = proj(f+*g,i).x
proof
  let f, g be non-empty Function, i, x being object;
  assume A1: x in product f /\ product(f+*g);
  then x in product f by XBOOLE_0:def 4;
  then reconsider h = x as Function;
  x in product f & x in product(f+*g) by A1, XBOOLE_0:def 4;
  then A2: x in dom proj(f,i) & x in dom proj(f+*g,i) by CARD_3:def 16;
  hence proj(f,i).x = h.i by CARD_3:def 16
    .= proj(f+*g,i).x by A2, CARD_3:def 16;
end;
