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 Th34:
  Lin Z_Lin A = 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: Lin(A) is Subspace of Lin(Z_Lin(A)) by RLVECT_3:20;
  reconsider W = the carrier of Lin(A) as Subset of V by RLSUB_1:def 2;
  Lin(Z_Lin A) is Subspace of Lin(W) by Th8,RLVECT_3:20; then
  Lin(Z_Lin A) is Subspace of Lin(A) by RLVECT_3:18;
  hence thesis by A1,RLSUB_1:26;
end;
