reserve x,y for set;
reserve a,b for Real;
reserve i,j for Integer;
reserve V for RealLinearSpace;
reserve W1,W2,W3 for Subspace of V;
reserve v,v1,v2,v3,u,w,w1,w2,w3 for VECTOR of V;
reserve A,B,C for Subset of V;
reserve L,L1,L2 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;

theorem
  for x be set holds x in Z_Lin(A) iff
    ex g1,h1 be FinSequence of V,
        a1 be INT-valued FinSequence
   st x=Sum(h1) &
       rng g1 c= A & len g1 =len h1 & len g1 = len a1 &
       for i be Nat
        st i in Seg (len g1) holds
           h1/.i=(a1.i)*(g1/.i) by Lm1,Lm2;
