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

theorem Th15:
for A be non empty closed_interval Subset of REAL,
    T0,T be DivSequence of A
  holds ex T1 be DivSequence of A st
   for i be Nat holds T1.(2*i) = T0.i & T1.(2*i+1) = T.i
proof
   let A be non empty closed_interval Subset of REAL,
       T0,T be DivSequence of A;
A1:dom T0 = NAT & dom T = NAT by FUNCT_2:def 1;
   now let i be object;
    assume i in NAT; then
    reconsider i1=i as Nat;
    rng (T0.i1) c= REAL;
    hence T0.i in (REAL)* by FINSEQ_1:def 11;
   end; then
   reconsider S0=T0 as sequence of (REAL)* by A1,FUNCT_2:3;
   now let i be object;
    assume i in NAT; then
    reconsider i1=i as Nat;
    rng (T.i1) c= REAL;
    hence T.i in (REAL)* by FINSEQ_1:def 11;
   end; then
   reconsider S=T as sequence of (REAL)* by A1,FUNCT_2:3;
   deffunc F(Nat) = S0/.$1;
   deffunc G(Nat) = S/.$1;
   consider T1 being sequence of (REAL)* such that
A2: for n be Nat holds T1.(2*n) = F(n) & T1.(2*n+1) = G(n) from ExRealSeq2X;
A3:dom T1 = NAT by FUNCT_2:def 1;
   now let i be object;
    assume i in NAT; then
    reconsider j=i as Nat;
    consider k be Nat such that
A4:  j=2*k or j=2*k+1 by Th14;
    reconsider k as Element of NAT by ORDINAL1:def 12;
    per cases by A4;
    suppose j=2*k;
     then T1.j = S0/.k by A2 .= T0.k;
     hence T1.i in divs A by INTEGRA1:def 3;
    end;
    suppose j=2*k + 1;
     then T1.j = S/.k by A2 .= T.k;
     hence T1.i in divs A by INTEGRA1:def 3;
    end;
   end; then
   reconsider T1 as DivSequence of A by A3,FUNCT_2:3;
   take T1;
   let i be Nat;
   i in NAT by ORDINAL1:def 12; then
A5:i in dom S0 & i in dom S by FUNCT_2:def 1;
A6:T1.(2*i) = S0/.i by A2 .= T0.i by A5,PARTFUN1:def 6;
   T1.(2*i+1) = S/.i by A2 .= T.i by A5,PARTFUN1:def 6;
   hence thesis by A6;
end;
