reserve R for Ring,
  V for RightMod of R,
  a,b for Scalar of R,
  x,y for set,
  p,q ,r for FinSequence,
  i,k for Nat,
  u,v,v1,v2,v3,w for Vector of V,
  F,G,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, R,
  S,T for finite Subset of V;

theorem Th9:
  T misses S implies Sum(T \/ S) = Sum(T) + Sum(S)
proof
  consider F such that
A1: rng F = T \/ S and
A2: F is one-to-one & Sum(T \/ S) = Sum(F) by Def1;
  consider G such that
A3: rng G = T and
A4: G is one-to-one and
A5: Sum(T) = Sum(G) by Def1;
  consider H such that
A6: rng H = S and
A7: H is one-to-one and
A8: Sum(S) = Sum(H) by Def1;
  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;
