reserve a,a1,b,b1,x,y for Real,
  F,G,H for FinSequence of REAL,
  i,j,k,n,m for Element of NAT,
  I for Subset of REAL,
  X for non empty set,
  x1,R,s for set;
reserve A for non empty closed_interval Subset of REAL;
reserve A, B for non empty closed_interval Subset of REAL;
reserve r for Real;
reserve D, D1, D2 for Division of A;
reserve f, g for Function of A,REAL;

theorem
  len F = len G & len F = len H & (for k st k in dom F holds H.k = F/.k
  + G/.k) implies Sum(H) = Sum(F) + Sum(G)
proof
  assume that
A1: len F = len G and
A2: len F = len H and
A3: for k st k in dom F holds H.k = F/.k + G/.k;
A4: F is Element of (len F)-tuples_on REAL by FINSEQ_2:92;
A5: G is Element of (len F)-tuples_on REAL by A1,FINSEQ_2:92;
  then F+G is Element of (len F)-tuples_on REAL by A4,FINSEQ_2:120;
  then
A6: len H = len (F+G) by A2,CARD_1:def 7;
  then
A7: dom H = Seg len(F+G) by FINSEQ_1:def 3;
A8: for k st k in dom F holds H.k = F.k + G.k
  proof
    let k;
    assume
A9: k in dom F;
    then k in Seg(len G) by A1,FINSEQ_1:def 3;
    then k in dom G by FINSEQ_1:def 3;
    then
A10: G/.k = G.k by PARTFUN1:def 6;
    F/.k = F.k by A9,PARTFUN1:def 6;
    hence thesis by A3,A9,A10;
  end;
  for k being Nat st k in dom H holds H.k = (F+G).k
  proof
    let k be Nat;
    assume
A11: k in dom H;
    then k in dom F by A2,A6,A7,FINSEQ_1:def 3;
    then
A12: H.k=F.k+G.k by A8;
    k in dom(F+G) by A7,A11,FINSEQ_1:def 3;
    hence thesis by A12,VALUED_1:def 1;
  end;
  then Sum H=Sum(F+G) by A6,FINSEQ_2:9
    .=Sum F+Sum G by A4,A5,RVSUM_1:89;
  hence thesis;
end;
