reserve i, j, m, n for Nat,
  z, B0 for set,
  f, x0 for real-valued FinSequence;

theorem Th8:
  for V being RealLinearSpace, X being Subspace of V st V is
  strict & X is strict & the carrier of X = the carrier of V holds X = V
proof
  let V be RealLinearSpace, X be Subspace of V;
  assume that
A1: V is strict and
A2: X is strict and
A3: the carrier of X=the carrier of V;
A4: the Mult of X = (the Mult of V) | ( [:REAL,the carrier of V:] ) by A3,
RLSUB_1:def 2
    .= the Mult of V;
A5: 0. X = 0. V by RLSUB_1:def 2;
  the addF of X = (the addF of V) || (the carrier of V) by A3,RLSUB_1:def 2
    .= (the addF of V) | ( [:the carrier of V,the carrier of V:] ) by
REALSET1:def 2
    .=the addF of V;
  hence X=V by A1,A2,A3,A5,A4;
end;
