
theorem Th4:
  for V be RealLinearSpace, V1 be Subset of V st
  V1 is linearly-closed non empty holds
  RLSStruct (# V1,Zero_(V1,V), Add_(V1,V),Mult_(V1,V) #) is Subspace of V
proof
  let V be RealLinearSpace;
  let V1 be Subset of V such that
A1: V1 is linearly-closed non empty;
A2: Mult_(V1,V) = (the Mult of V) | [:REAL,V1:] by A1,Def6;
  Zero_(V1,V) = 0.V & Add_(V1,V)= (the addF of V)||V1 by A1,Def5,Def7;
  hence thesis by A1,A2,RLSUB_1:24;
end;
