
theorem Th15:
  for T be FinSequence of REAL, n,m be Nat
  st n+1 < m <= len T holds
    ex TM1 be 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 FinSequence of REAL, n,m be Nat;
    assume
    A1: n+1 < m <= len T;
    deffunc F(Nat) = T.($1+n);
    m-(n+1) in NAT by A1,INT_1:5; then
    reconsider m1 = m-(n+1) as Nat;
    consider p being FinSequence such that
    A2: len p = m1
      & for k being Nat st k in dom p holds p.k = F(k) from FINSEQ_1:sch 2;
    A3: rng p c= rng T
    proof
      let x be object;
      assume x in rng p; then
      consider i be object such that
      A4: i in dom p & x = p.i by FUNCT_1:def 3;
      reconsider i as Nat by A4;
      A6: p.i = T.(i+n) by A2,A4;
      A7: 1 <= i <= m1 by A2,FINSEQ_3:25,A4;
      A8: i + n <= m1 + n by A7,XREAL_1:6;
      m - 1 <= m - 0 by XREAL_1:10; then
      m - 1 <= len T by A1,XXREAL_0:2; then
      A9: i + n <= len T by A8,XXREAL_0:2;
      1 + 0 <= i + n by A7,XREAL_1:7; then
      i+n in dom T by FINSEQ_3:25,A9;
      hence thesis by A4,A6,FUNCT_1:3;
    end; then
    reconsider p as FinSequence of REAL by FINSEQ_1:def 4,XBOOLE_1:1;
    take p;
    thus len p = m-(n+1) by A2;
    thus rng p c= rng T by A3;
    let i be Nat;
    assume i in dom p;
    hence thesis by A2;
  end;
