
theorem RngForArProg:
  rng (ArProg(199,210) | 10) =
    {199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089}
  proof
    set f = ArProg(199,210) | 10;
    set g = ArProg(199,210);
    10 c= NAT; then
B4: 10 c= dom g by FUNCT_2:def 1;
B0: dom f = dom g /\ 10 by RELAT_1:61
         .= 10 by B4,XBOOLE_1:28;
    thus rng (ArProg(199,210) | 10) c=
      {199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089}
    proof
      let x be object;
      assume x in rng (ArProg(199,210) | 10); then
      consider y being object such that
B1:   y in dom f & x = f.y by FUNCT_1:def 3;
B3:   x = g.y by B0,B1,FUNCT_1:49;
      y = 0 or y = 1 or y = 2 or y = 3 or y = 4 or y = 5 or
        y = 6 or y = 7 or y = 8 or y = 9
          by CARD_1:58,B0,B1,ENUMSET1:def 8;
      hence thesis by ENUMSET1:def 8,B3,Th1990,Th1991,Th1992,Th1993,
        Th1994,Th1995,Th1996,Th1997,Th1998,Th1999;
    end;
    let x be object;
    assume x in {199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089};
      then
HJ: x = g.0 or x = g.1 or x = g.2 or x = g.3 or x = g.4 or x = g.5 or
      x = g.6 or x = g.7 or x = g.8 or x = g.9
        by Th1990,Th1991,Th1992,Th1993,Th1994,Th1995,Th1996,Th1997,
          Th1998,Th1999,ENUMSET1:def 8;
    x in rng (ArProg(199,210) | 10)
    proof
J3:   0 in dom f & 1 in dom f & 2 in dom f & 3 in dom f & 4 in dom f &
      5 in dom f & 6 in dom f & 7 in dom f & 8 in dom f & 9 in dom f
        by CARD_1:58,ENUMSET1:def 8,B0; then
      f.0 = g.0 & f.1 = g.1 & f.2 = g.2 & f.3 = g.3 & f.4 = g.4 &
        f.5 = g.5 & f.6 = g.6 & f.7 = g.7 & f.8 = g.8 & f.9 = g.9
          by FUNCT_1:47;
      hence thesis by J3,FUNCT_1:3,HJ;
    end;
    hence thesis;
  end;
