
theorem Th6:
  for V be ComplexLinearSpace, V1 be Subset of V
    st V1 is linearly-closed non empty
  holds CLSStruct (# V1,Zero_(V1,V), Add_(V1,V), Mult_(V1,V) #)
   is Subspace of V
proof
  let V be ComplexLinearSpace;
  let V1 be Subset of V such that
A1: V1 is linearly-closed and
A2: V1 is non empty;
A3: Add_(V1,V)= (the addF of V) || V1 by A1,Def6;
A4: Mult_(V1,V) = (the Mult of V) | [:COMPLEX,V1:] by A1,Def7;
  Zero_(V1,V) = 0.V by A1,A2,Def8;
  hence thesis by A2,A3,A4,CLVECT_1:43;
end;
