
theorem Th15:
  for X be RealLinearSpace, x be sequence of X, A be finite Subset of X
  st A c= rng x
  holds ex n be Nat st A c= rng ( x | Segm n )
  proof
    let X be RealLinearSpace, x be sequence of X;
    defpred P[Nat] means
    for A be finite Subset of X st card A = $1 & A c= rng x
    holds ex n be Nat st A c= rng (x | Segm n);
    A1: P[0]
    proof
      let A be finite Subset of the carrier of X;
      assume
      A2: card A = 0 & A c= rng x;
      set n = the Nat;
      take n;
      thus A c= rng (x | Segm n) by A2;
    end;
    A3: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
      A4: P[k];
      let A be finite Subset of the carrier of X;
      assume
      A5: card A = k+1 & A c= rng x; then
      A <> {}; then
      consider w be object such that
      A6: w in A by XBOOLE_0:def 1;
      reconsider w as Element of the carrier of X by A6;
      A7: card(A \ {w}) = card(A) - card({w}) by A6,CARD_2:44,ZFMISC_1:31
                       .= k + 1 - 1 by A5,CARD_2:42;
      reconsider A0 =A \ {w} as finite Subset of X;
      A0 c= A by XBOOLE_1:36; then
      A0 c= rng x by A5; then
      consider n0 be Nat such that
      A8: A0 c= rng ( x | Segm n0 ) by A4,A7;
      consider n1 be object such that
      A9: n1 in NAT & w = x.n1 by A5,A6,FUNCT_2:11;
      reconsider n1 as Nat by A9;
      set n = n0 + n1 + 1;
      take n;
      A10: A = A0 \/ {w} by A6,XBOOLE_1:45,ZFMISC_1:31;
      n0 < n by NAT_1:11,13; then
      x | Segm n0 c= x | Segm n by NAT_1:39,RELAT_1:75; then
      rng ( x | Segm n0 ) c= rng ( x | Segm n ) by RELAT_1:11; then
      A11: A0 c= rng ( x | Segm n ) by A8;
      n1 < n by NAT_1:11,13; then
      n1 in Segm n & dom x = NAT by FUNCT_2:def 1,NAT_1:44; then
      {w} c= rng ( x | Segm n ) by A9,FUNCT_1:50,ZFMISC_1:31;
      hence A c= rng ( x | Segm n ) by A10,A11,XBOOLE_1:8;
    end;
    A12: for k be Nat holds P[k] from NAT_1:sch 2(A1,A3);
    let A be finite Subset of the carrier of X;
    assume
    A13: A c= rng x;
    card A is Nat;
    hence ex n be Nat st A c= rng ( x | Segm n ) by A12,A13;
  end;
