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

theorem
  for A be FinSequence of n-tuples_on BOOLEAN,
  C be Subset of n-BinaryVectSp
  st 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)) &
  C c= rng A
  holds Lin C is Subspace of n-BinaryVectSp & C is Basis of Lin C &
  dim (Lin C) = card C
proof
  let A be FinSequence of n-tuples_on BOOLEAN,
  C be Subset of n-BinaryVectSp;
  assume
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));
  assume
A2: C c= rng A;
  reconsider B = rng A as finite Subset of n-BinaryVectSp;
  B is linearly-independent by A1,Th10;
  then
A3: C is linearly-independent by A2,VECTSP_7:1;
  for x be object st x in C holds x in the carrier of Lin C
  by VECTSP_7:8,STRUCT_0:def 5;
  then reconsider C0 = C as Subset of (Lin C) by TARSKI:def 3;
  Lin C0 = the ModuleStr of (Lin C) by VECTSP_9:17;
  then C0 is Basis of Lin C by VECTSP_7:def 3,A3,VECTSP_9:12;
  hence thesis by VECTSP_9:def 1;
end;
