reserve i,j,k,n,m for Nat,
  x,y,z,y1,y2 for object, X,Y,D for set,
  p,q for XFinSequence;
reserve k1,k2 for Nat;
reserve D for non empty set,
  F,G for XFinSequence of D,
  b for BinOp of D,
  d,d1,d2 for Element of D;
reserve F for XFinSequence,
        rF,rF1,rF2 for real-valued XFinSequence,
        r for Real,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;

theorem :: NUMERAL1:2
  for S being Real_Sequence st rF=S|(n+1) holds Sum rF = Partial_Sums(S).n
proof
  let S be Real_Sequence;
  n+1 c= NAT;
  then
A2: n+1 c= dom S by FUNCT_2:def 1;
  assume
A3: rF=S|(n+1);
  then dom rF = dom S /\ (n+1) by RELAT_1:61;
  then
A4: dom rF = n+1 by A2,XBOOLE_1:28;
  then consider f be sequence of REAL such that
A5: f.0 = rF.0 and
A6: for m be Nat st m+1 < len rF holds f.(m + 1) = addreal.(f.m,rF.(m + 1)) and
A7: addreal "**" rF = f.(len rF-1) by Def8;
  defpred P[Nat] means $1 in dom rF implies f.$1=Partial_Sums(S).$1;
A8: now
    let k;
    assume
A9: P[k];
    thus P[k+1]
    proof
      assume
A10:  k+1 in dom rF;
      then
A11:  k+1 < len rF by AFINSQ_1:86;
      then
A12:  k<len rF by NAT_1:13;
      thus f.(k+1)= addreal.(f.k,rF.(k + 1)) by A6,A11
        .= (f.k)+rF.(k + 1) by BINOP_2:def 9
        .= (f.k)+S.(k+1) by A3,A10,FUNCT_1:47
        .= Partial_Sums(S).(k+1) by A9,A12,AFINSQ_1:86,SERIES_1:def 1;
    end;
  end;
  Partial_Sums(S).0=S.0 by SERIES_1:def 1;
  then
A13: P[0] by A3,A5,FUNCT_1:47;
A14: n in Segm(n+1) by NAT_1:45;
  for m holds P[m] from NAT_1:sch 2(A13,A8);
  hence Partial_Sums(S).n=f.n by A4,A14
     .= Sum rF by Th47,A7,A4;
end;
