
theorem Th19:
  for f being trivial Function, x being object st x in dom f
  holds proj(f,x) is one-to-one
proof
  let f be trivial Function, x be object;
  assume A1: x in dom f;
  then consider t being object such that
    A2: dom f = {t} by ZFMISC_1:131;
  A3: dom f = {x} by A1, A2, TARSKI:def 1;
  set F = proj(f,x);
  for y, z being object st y in dom F & z in dom F & F.y = F.z holds y = z
  proof
    let y,z be object;
    assume A4: y in dom F & z in dom F & F.y = F.z;
    then consider g being Function such that
      A5: y = g & dom g = dom f and
      for s being object st s in dom f holds g.s in f.s
        by CARD_3:def 5;
    consider h being Function such that
      A6: z = h & dom h = dom f and
      for s being object st s in dom f holds h.s in f.s
        by A4, CARD_3:def 5;
    A7: g.x = F.z by A4, A5, CARD_3:def 16
      .= h.x by A4, A6, CARD_3:def 16;
    for s being object st s in dom g holds g.s = h.s
    proof
      let s be object;
      assume s in dom g;
      then s = x by A3, A5, TARSKI:def 1;
      hence thesis by A7;
    end;
    hence thesis by A5, A6, FUNCT_1:2;
  end;
  hence thesis by FUNCT_1:def 4;
end;
