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

theorem Th10:
  for A be FinSequence of n-tuples_on BOOLEAN,
  B be finite Subset of n-BinaryVectSp
  st rng A = B & m <= n & len A = m & A is one-to-one &
  (for i,j be Nat st i in Seg n & j in Seg m holds
  (i = j implies (A.i).j = TRUE) & (i <> j implies (A.i).j = FALSE))
  holds B is linearly-independent
proof
  let A be FinSequence of n-tuples_on BOOLEAN,
  B be finite Subset of n-BinaryVectSp;
  assume that
A1: rng A = B and
A2: m <= n and
A3: len A = m and
A4: A is one-to-one and
A5: for i,j be Nat st i in Seg n & j in Seg m holds
  (i = j implies (A.i).j = TRUE) & (i <> j implies (A.i).j = FALSE);
  set V = n-BinaryVectSp;
  for l being Linear_Combination of B st Sum l = 0.V
  holds Carrier l = {}
  proof
    let l be Linear_Combination of B;
    assume
A6: Sum l = 0.V;
    assume Carrier l <> {};
    then consider x be object such that
A7: x in Carrier l by XBOOLE_0:def 1;
    consider v be Element of V such that
A8: x = v & l.v <> 0.Z_2 by A7;
A9: Carrier l c= B by VECTSP_6:def 4;
    reconsider Suml = Sum l as Element of n-tuples_on BOOLEAN;
    consider j be object such that
A10: j in dom A & v = A.j by A1,A9,A7,A8,FUNCT_1:def 3;
A11: j in Seg m by A3,A10,FINSEQ_1:def 3;
    reconsider j as Nat by A10;
    (Suml).j = l.(A.j) by Th9,A1,A2,A3,A4,A5,A11;
    hence contradiction by A6,A8,A10,BSPACE:5;
  end;
  hence thesis by VECTSP_7:def 1;
end;
