
theorem Th3:
for V being RealLinearSpace,
    A,B being Subset of V
    st A c= the carrier of Lin B
     & B c= the carrier of Lin A holds
    Lin A = Lin B
proof
  let V be RealLinearSpace,
      A,B be Subset of V;
  assume that
  A1: A c= the carrier of Lin B and
  A2: B c= the carrier of Lin A;
A3: Lin A is Subspace of Lin B by A1,Th2;
  Lin B is Subspace of Lin A by A2,Th2;
  hence thesis by RLSUB_1:26,A3;
end;
