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(A) = Z_Lin(B) implies Lin(A) = Lin(B)
proof
  assume Z_Lin(A) = Z_Lin(B);
  hence Lin(A) = Lin( Z_Lin(B)) by Th34
            .= Lin(B) by Th34;
end;
