theorem Th20:
  A c= B implies Lin(A) is Subspace of Lin(B)
proof
  assume
A1: A c= B;
  now
    let v;
    assume v in Lin(A);
    then consider l such that
A2: v = Sum(l) by Th14;
    reconsider l as Linear_Combination of B by A1,RLVECT_2:21;
    Sum(l) = v by A2;
    hence v in Lin(B) by Th14;
  end;
  hence thesis by RLSUB_1:29;
end;
