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 Th9:
  v in Z_Lin(A) & u in Z_Lin(A) implies v + u in Z_Lin(A)
  proof
    assume that
A1: v in Z_Lin(A) and
A2: u in Z_Lin(A);
    consider l1 being Linear_Combination of A such that
A3: v = Sum(l1) & rng l1 c= INT by A1;
    consider l2 being Linear_Combination of A such that
A4: u = Sum(l2) & rng l2 c= INT by A2;
    reconsider f = l1 + l2 as Linear_Combination of A by RLVECT_2:38;
A5: rng f c= INT by A3,A4,Th5;
    v + u = Sum(f) by A3,A4,RLVECT_3:1;
    hence thesis by A5;
  end;
