reserve n for Nat,
        p,p1,p2 for Point of TOP-REAL n,
        x for Real;
reserve n,m for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS m,REAL-NS n;
reserve g for PartFunc of REAL m,REAL n;
reserve h for PartFunc of REAL m,REAL;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL m;
reserve X for set;

theorem Th27:
for n be non zero Nat, h,g,j be FinSequence of REAL n st
 len h= len j & len g= len j
 & (for i be Nat st i in dom j holds j/.i = h /.i - g/.i )
holds Sum j = Sum h - Sum g
proof
   let n be non zero Nat,
       h,g,j be FinSequence of REAL n;
   assume that
A1: len h = len j & len g = len j and
A2: for i be Nat st i in dom j holds j/.i = h/.i - g/.i;
A3:
dom j = Seg len j & dom g = Seg len g & dom h = Seg len h by FINSEQ_1:def 3;
A4:for i be Nat st i in dom h holds h/.i = j/.i + g/.i
   proof
    let i be Nat;
    reconsider ji = j/.i, hi = h/.i, gi = g/.i as Point of TOP-REAL n
       by EUCLID:22;
    assume i in dom h;
    then ji = hi - gi by A1,A2,A3;
    then ji + gi = hi by RLVECT_4:1;
    hence thesis;
   end;
   j+g = j<++>g by INTEGR15:def 9;
   then
A5:dom (j+g) = dom j /\ dom g by VALUED_2:def 45;
   reconsider Sj = Sum j, Sh = Sum h, Sg = Sum g as Point of TOP-REAL n
       by EUCLID:22;
    for k being Element of NAT st k in dom h holds h.k = (j+g).k
   proof
    let k be Element of NAT;
    assume A6: k in dom h;
    then h/.k = j/.k + g/.k by A4;
then A7: h.k = j/.k + g/.k by A6,PARTFUN1:def 6;
     (j+g)/.k = j/.k + g/.k by A6,A1,A3,A5,INTEGR15:21;
    hence thesis by A7,A6,A1,A3,A5,PARTFUN1:def 6;
   end;
   then Sh = Sj + Sg by A1,A3,Th24,A5,PARTFUN1:5;
   then Sh - Sg = Sj by RLVECT_4:1;
   hence Sum h - Sum g = Sum j;
end;
