
theorem Th22:
  for Sq0,Sq,Sq1 be Real_Sequence
  st Sq1 is convergent
   & for i be Nat holds Sq1.(2*i) = Sq0.i & Sq1.(2*i+1) = Sq.i holds
    Sq0 is convergent & lim Sq0 = lim Sq1
  & Sq is convergent & lim Sq = lim Sq1
  proof
    let Sq0,Sq,Sq1 be Real_Sequence;
    assume that
    A1: Sq1 is convergent and
    A2: for i be Nat holds Sq1.(2*i) = Sq0.i & Sq1.(2*i+1) = Sq.i;
    A3: for r be Real st 0 < r
        ex m1 be Nat st for i be Nat st m1 <= i holds |. Sq0.i - lim Sq1 .| <r
    proof
      let r be Real;
      assume 0 < r; then
      consider m be Nat such that
      A4: for n be Nat st m <= n holds
          |. Sq1.n - lim Sq1 .|<r by A1,SEQ_2:def 7;
      consider k be Nat such that
      A5: m = 2*k or m = 2*k+1 by INTEGR20:14;
      2*k+1 <= 2*k+1+1 by NAT_1:11; then
      A6:m <= 2*k+2 by A5,XREAL_1:31;
      reconsider m1 = k+1 as Nat;
      take m1;
      thus for i be Nat st m1 <= i holds |. Sq0.i - lim Sq1 .| < r
      proof
        let i be Nat;
        assume m1 <= i; then
        2*m1 <= 2*i by XREAL_1:64; then
        m <= 2*i by A6,XXREAL_0:2; then
        |. Sq1.(2*i) - lim Sq1 .| < r by A4;
        hence |. Sq0.i - lim Sq1 .| < r by A2;
      end;
    end;
    hence Sq0 is convergent by SEQ_2:def 6;
    hence lim Sq0 = lim Sq1 by A3,SEQ_2:def 7;
    A7: for r be Real st 0 < r
        ex m1 be Nat st for i be Nat
        st m1 <= i holds |. Sq.i - lim Sq1 .| < r
    proof
      let r be Real;
      assume 0 < r; then
      consider m be Nat such that
      A8: for n be Nat st m <= n holds
          |. Sq1.n - lim Sq1 .| < r by A1,SEQ_2:def 7;
      consider k be Nat such that
      A9: m = 2*k or m = 2*k+1 by INTEGR20:14;
      2*k+1 <= 2*k+1+1 by NAT_1:11; then
      A10: m <= 2*k+2 by A9,XREAL_1:31;
      reconsider m1 = k+1 as Nat;
      take m1;
      thus for i be Nat st m1 <= i holds |. Sq.i - lim Sq1 .| < r
      proof
        let i be Nat;
        assume m1 <= i; then
        2*m1 <= 2*i by XREAL_1:64; then
        m <= 2*i by A10,XXREAL_0:2; then
        m <= 2*i+1 by XREAL_1:145; then
        |. Sq1.(2*i+1) - lim Sq1 .| < r by A8;
        hence |. Sq.i - lim Sq1 .| < r by A2;
      end;
    end;
    hence Sq is convergent by SEQ_2:def 6;
    hence lim Sq = lim Sq1 by SEQ_2:def 7,A7;
  end;
