reserve x,y for object, X,Y,Z for set;
reserve a,b for Real;
reserve k for Element of NAT;
reserve V for RealLinearSpace;
reserve W1,W2,W3 for Subspace of V;
reserve v,v1,v2,u for VECTOR of V;
reserve A,B,C for Subset of V;
reserve T for finite Subset of V;
reserve L,L1,L2 for Linear_Combination of V;
reserve l for Linear_Combination of A;
reserve F,G,H for FinSequence of the carrier of V;
reserve f,g for Function of the carrier of V, REAL;
reserve p,q,r for FinSequence;
reserve M for non empty set;
reserve CF for Choice_Function of M;

theorem Th4:
  Sum(L1 - L2) = Sum(L1) - Sum(L2)
proof
  thus Sum(L1 - L2) = Sum(L1) + Sum(- L2) by Th1
    .= Sum(L1) + (- Sum(L2)) by Th3
    .= Sum(L1) - Sum(L2) by RLVECT_1:def 11;
end;
