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 Th10:
  v in Z_Lin(A) implies i*v in Z_Lin(A)
  proof
    assume v in Z_Lin(A);
    then consider l such that
A1: v = Sum(l) & rng l c= INT;
    reconsider a=i as Real;
A2: a*l=i*l by Th4; then
    reconsider f = i * l as Linear_Combination of A by RLVECT_2:44;
A3: i * v = Sum(f) by A1,A2,RLVECT_3:2;
    rng (i * l) c= INT by Th6,A1;
    hence thesis by A3;
  end;
