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;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;
reserve RR for domRing;
reserve VV for RightMod of RR;
reserve LL for Linear_Combination of VV;
reserve aa for Scalar of RR;
reserve uu, vv for Vector of VV;

theorem
  L1 is Linear_Combination of A & L2 is Linear_Combination of A implies
  L1 - L2 is Linear_Combination of A
proof
  assume that
A1: L1 is Linear_Combination of A and
A2: L2 is Linear_Combination of A;
  - L2 is Linear_Combination of A by A2,Th45;
  hence thesis by A1,Th38;
end;
