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 Th13:
  A c= B implies Z_Lin(A) c= Z_Lin(B)
proof
  assume
A1: A c= B;
  let v be object;
    assume v in Z_Lin(A);
    then consider l such that
A2: v = Sum(l) & rng l c= INT;
    reconsider l as Linear_Combination of B by A1,RLVECT_2:21;
    Sum(l) = v by A2;
    hence v in Z_Lin(B) by A2;
end;
