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

theorem Th17:
for X be RealNormSpace, A be non empty closed_interval Subset of REAL,
    h be Function of A,the carrier of X, T0,T,T1 be DivSequence of A,
    S0 be middle_volume_Sequence of h,T0,
    S be middle_volume_Sequence of h,T
  st for i be Nat holds T1.(2*i) = T0.i & T1.(2*i+1) = T.i
 ex S1 be middle_volume_Sequence of h,T1
   st for i be Nat holds S1.(2*i) = S0.i & S1.(2*i+1) = S.i
proof
   let X be RealNormSpace, A be non empty closed_interval Subset of REAL,
       h be Function of A,the carrier of X, T0,T,T1 be DivSequence of A,
       S0 be middle_volume_Sequence of h,T0,
       S be middle_volume_Sequence of h,T;
   assume A1: for k be Nat holds T1.(2*k) = T0.k & T1.(2*k+1) = T.k;
   reconsider SS0=S0, SS=S as sequence of (the carrier of X)*;
   deffunc F(Nat) = SS0/.$1;
   deffunc G(Nat) = SS/.$1;
   consider S1 being sequence of (the carrier of X)* such that
A2: for n be Nat holds S1.(2*n) = F(n) & S1.(2*n+1) = G(n)
     from ExRealSeq2X;
   for i be Element of NAT holds S1.i is middle_volume of h,T1.i
   proof
    let i be Element of NAT;
    consider k be Nat such that
A3:   i=2*k or i=2*k+1 by Th14;
    reconsider k as Element of NAT by ORDINAL1:def 12;
    per cases by A3;
    suppose A4: i=2*k;
     then S1.i = SS0/.k by A2 .= S0.k;
     hence S1.i is middle_volume of h,T1.i by A4,A1;
    end;
    suppose A5: i=2*k + 1; then
     S1.i = SS/.k by A2 .= S.k;
     hence S1.i is middle_volume of h,T1.i by A5,A1;
    end;
   end;
   then reconsider S1 as middle_volume_Sequence of h,T1 by INTEGR18:def 3;
   take S1;
   let i be Nat;
   i in NAT by ORDINAL1:def 12; then
A6:i in dom SS0 & i in dom SS by FUNCT_2:def 1;
A7:S1.(2*i) = SS0/.i by A2 .= S0.i by PARTFUN1:def 6,A6;
   S1.(2*i+1) = SS/.i by A2 .= S.i by PARTFUN1:def 6,A6;
   hence thesis by A7;
end;
