 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem Th36:
  for f be summable Real_Sequence, n be Nat st
    f is positive-yielding holds
      Sum (f ^\ (n+1)) > 0
  proof
    let f be summable Real_Sequence, n be Nat;
    assume
A0: f is positive-yielding;
    set LS = f ^\ (n + 1);
A2: for i be Nat holds 0 <= LS.i
    proof
      let i be Nat;
a1:   LS.i = f.(n+1+i) by NAT_1:def 3;
      n+1+i in NAT by ORDINAL1:def 12; then
      n+1+i in dom f by FUNCT_2:def 1; then
      f.(n+1+i) in rng f by FUNCT_1:3;
      hence thesis by PARTFUN3:def 1,A0,a1;
    end;
    ex i be Nat st i in dom LS & 0 < LS.i
    proof
      consider j be Nat such that
A3:   n + 1 <= j;
      j - (n + 1) in NAT by A3,INT_1:5; then
      reconsider i = j - (n + 1) as Nat;
      take i;
A4:   n + 1 + i = j;
aa:   dom LS = NAT by FUNCT_2:def 1;
A6:   LS.i = f.j by NAT_1:def 3,A4;
      j in NAT by ORDINAL1:def 12; then
      j in dom f by FUNCT_2:def 1; then
      f.j in rng f by FUNCT_1:3;
      hence thesis by aa,A6,A0,PARTFUN3:def 1,ORDINAL1:def 12;
    end; then
    consider k be Nat such that
A6: k in dom LS & LS.k > 0;
    LS is summable by SERIES_1:12;
    hence thesis by A6,RSSPACE2:3,A2;
  end;
