reserve m,n,s for non zero Element of NAT;

theorem Th13:
  ex B be finite Subset of n-BinaryVectSp
  st B is Basis of n-BinaryVectSp & card B = n &
  ex A be FinSequence of n-tuples_on BOOLEAN st len A = n &
  A is one-to-one & card (rng A) = n & rng A = B &
  (for i,j be Nat st i in Seg n & j in Seg n holds
  (i = j implies (A.i).j = TRUE) & (i <> j implies (A.i).j = FALSE))
proof
  set V = n-BinaryVectSp;
  consider A be FinSequence of n-tuples_on BOOLEAN such that
A1: len A = n & A is one-to-one & card (rng A) = n &
  (for i,j be Nat st i in Seg n & j in Seg n holds
  (i = j implies (A.i).j = TRUE) & (i <> j implies (A.i).j = FALSE))
  by Th8;
  reconsider B = rng A as finite Subset of n-BinaryVectSp;
A2: B is linearly-independent by A1,Th10;
  Lin B = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V,
  the lmult of V #)  by A1,Th12;
  then B is Basis of V by VECTSP_7:def 3,A2;
  hence thesis by A1;
end;
