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

theorem Th18:
for X be RealNormSpace, Sq0,Sq,Sq1 be sequence of X
  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 X be RealNormSpace, Sq0,Sq,Sq1 be sequence of X;
   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,NORMSP_1:def 7;
    consider k be Nat such that
A5:   m=2*k or m=2*k+1 by Th14;
    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;
   hence lim Sq0=lim Sq1 by A3,NORMSP_1: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,NORMSP_1:def 7;
    consider k be Nat such that
A9:   m=2*k or m=2*k+1 by Th14;
    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;
   hence lim Sq=lim Sq1 by NORMSP_1:def 7,A7;
end;
