reserve s1,s2,q1 for Real_Sequence;
reserve n for Element of NAT;
reserve a,b for Real;

theorem Th16:
for A be non empty closed_interval Subset of REAL,
    T0,T,T1 be DivSequence of A
  st delta T0 is convergent & lim delta T0 = 0
   & delta T is convergent & lim delta T = 0
   & for i be Nat holds T1.(2*i) = T0.i & T1.(2*i+1) = T.i
holds delta T1 is convergent & lim delta T1 = 0
proof
   let A be non empty closed_interval Subset of REAL,
       T0,T,T1 be DivSequence of A;
   assume that
A1: delta T0 is convergent & lim delta T0 = 0 and
A2: delta T is convergent & lim delta T = 0 and
A3: for i be Nat holds T1.(2*i) = T0.i & T1.(2*i+1) = T.i;
A4:now let p be Real;
    assume A5: 0<p; then
    consider n1 be Nat such that
A6:  for m be Nat st n1 <= m holds |. (delta T0).m - 0 .|<p by A1,SEQ_2:def 7;
    consider n2 be Nat such that
A7:  for m be Nat st n2 <= m holds |. (delta T).m - 0 .|<p
       by A5,A2,SEQ_2:def 7;
    reconsider n3=max(n1,n2) as Nat by TARSKI:1;
A8: n1 <= n3 & n2 <= n3 by XXREAL_0:25;
    reconsider n=2*n3+1 as Nat;
    take n;
    let m be Nat;
    assume A9: n <= m;
    consider k be Nat such that
A10:   m=2*k or m=2*k+1 by Th14;
    reconsider k1=k, m1=m as Element of NAT by ORDINAL1:def 12;
    per cases by A10;
    suppose A11: m=2*k;
     (delta T1).m1 = delta(T1.m1) by INTEGRA3:def 2; then
     (delta T1).m1 = delta(T0.k1) by A11,A3; then
A12: (delta T1).m1 = (delta T0).k1 by INTEGRA3:def 2;
A13:  2*n3 <= (2*k)-1 by XREAL_1:19,A9,A11;
     2*k-1 < (2*k-1)+1 by XREAL_1:145; then
     2*n3 <= 2*k by A13,XXREAL_0:2; then
     2*n3/2 <= 2*k/2 by XREAL_1:72; then
     n1 <= k by A8,XXREAL_0:2;
     hence |. (delta T1).m - 0 .|<p by A12,A6;
    end;
    suppose A14: m=2*k+1;
     (delta T1).m1 = delta(T1.m1) by INTEGRA3:def 2; then
     (delta T1).m1 = delta(T.k1) by A14,A3; then
A15: (delta T1).m1 = (delta T).k1 by INTEGRA3:def 2;
     (2*n3+1)-1 <= (2*k+1)-1 by XREAL_1:13,A9,A14; then
     2*n3/2 <= 2*k/2 by XREAL_1:72; then
     n2 <= k by A8,XXREAL_0:2;
     hence |. (delta T1).m - 0 .|<p by A15,A7;
    end;
   end;
   hence delta T1 is convergent by SEQ_2:def 6;
   hence lim delta T1 = 0 by A4,SEQ_2:def 7;
end;
