
theorem Th7:
  for V being RealUnitarySpace, A,B being Subset of V st A c= B
  holds Lin(A) is Subspace of Lin(B)
proof
  let V be RealUnitarySpace;
  let A,B be Subset of V;
  assume
A1: A c= B;
  now
    let v be VECTOR of V;
    assume v in Lin(A);
    then consider l being Linear_Combination of A such that
A2: v = Sum(l) by Th1;
    reconsider l as Linear_Combination of B by A1,RLVECT_2:21;
    Sum(l) = v by A2;
    hence v in Lin(B) by Th1;
  end;
  hence thesis by RUSUB_1:23;
end;
