
theorem Th16:
  for T be non empty increasing FinSequence of REAL, n,m be Nat
  st n+1 < m <= len T holds
    ex TM1 be non empty increasing FinSequence of REAL
    st len TM1 = m-(n+1) & rng TM1 c= rng T
     & for i be Nat st i in dom TM1 holds TM1.i = T.(i+n)
  proof
    let T be non empty increasing FinSequence of REAL, n,m be Nat;
    assume
    A1: n+1 < m <= len T; then
    consider p be FinSequence of REAL such that
    A2: len p = m-(n+1) & rng p c= rng T
      & for i be Nat st i in dom p holds p.i = T.(i+n) by Th15;
    m-(n+1) in NAT by A1,INT_1:5; then
    reconsider m1 = m-(n+1) as Nat;
    len p <> 0 by A1,A2; then
    reconsider p as non empty FinSequence of REAL;
    for e1,e2 be ExtReal st e1 in dom p & e2 in dom p & e1 < e2
    holds p.e1 < p.e2
    proof
      let e1,e2 be ExtReal;
      assume
      A3: e1 in dom p & e2 in dom p & e1 < e2; then
      reconsider ie1 = e1,ie2 = e2 as Nat;
      A5: p.e1 = T.(ie1+n) by A2,A3;
      A6: p.e2 = T.(ie2+n) by A2,A3;
      A7: 1 <= ie1 <= m1 by A2,A3,FINSEQ_3:25;
      A8: 1 <= ie2 <= m1 by A2,A3,FINSEQ_3:25;
      A9: ie1 + n <= m1 + n by A7,XREAL_1:6;
      m - 1 <= m - 0 by XREAL_1:10; then
      A10: m - 1 <= len T by A1,XXREAL_0:2; then
      A11: ie1+n <= len T by A9,XXREAL_0:2;
      1 + 0 <= ie1 + n by A7,XREAL_1:7; then
      A12: ie1+n in dom T by A11,FINSEQ_3:25;
      A13: 1 + 0 <= ie2 + n by A8,XREAL_1:7;
      ie2 + n <= m1 + n by A8,XREAL_1:6; then
      ie2 + n <= len T by A10,XXREAL_0:2; then
      A14: ie2 + n in dom T by FINSEQ_3:25,A13;
      ie1 + n < ie2 + n by A3,XREAL_1:8;
      hence thesis by A5,A6,A12,A14,VALUED_0:def 13;
    end; then
    reconsider p as non empty increasing FinSequence of REAL
      by VALUED_0:def 13;
    take p;
    thus thesis by A2;
  end;
