reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;

theorem Th12:
  for V be Abelian add-associative right_zeroed non empty
  addLoopStr, S,T be finite Subset of V holds T misses S implies Sum(T \/ S) =
  Sum(T) + Sum(S)
proof
  let V be Abelian add-associative right_zeroed non empty addLoopStr, S,T be
  finite Subset of V;
  consider F be FinSequence of the carrier of V such that
A1: rng F = T \/ S and
A2: F is one-to-one & Sum(T \/ S) = Sum(F) by Def2;
  consider G be FinSequence of the carrier of V such that
A3: rng G = T and
A4: G is one-to-one and
A5: Sum(T) = Sum(G) by Def2;
  consider H be FinSequence of the carrier of V such that
A6: rng H = S and
A7: H is one-to-one and
A8: Sum(S) = Sum(H) by Def2;
  set I = G ^ H;
  assume T misses S;
  then
A9: I is one-to-one by A3,A4,A6,A7,FINSEQ_3:91;
  rng I = rng F by A1,A3,A6,FINSEQ_1:31;
  hence Sum(T \/ S) = Sum(I) by A2,A9,RLVECT_1:42
    .= Sum(T) + Sum(S) by A5,A8,RLVECT_1:41;
end;
