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
  Z_Lin Z_Lin A = Z_Lin A
proof
  for x be object st x in A holds x in Z_Lin(A) by Th12;
  then A c= Z_Lin(A); then
A1: Z_Lin(A) c= Z_Lin( Z_Lin(A)) by Th13;
  Z_Lin Z_Lin A c= Z_Lin(A)
  proof
    let x be object;
    assume x in Z_Lin( Z_Lin(A));
    then consider g1,h1 be FinSequence of V,
    a1 be INT-valued FinSequence such that
A2: x=Sum(h1) &
    rng g1 c= Z_Lin(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 Lm2;
    thus x in Z_Lin(A) by A2,Th35;
  end;
  hence thesis by A1,XBOOLE_0:def 10;
end;
