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

theorem Th11:
  for A be FinSequence of n-tuples_on BOOLEAN,
  B be finite Subset of n-BinaryVectSp,
  v be Element of n-tuples_on BOOLEAN
  st rng A = B & len A = n & A is one-to-one
  ex l being Linear_Combination of B st
  for j be Nat st j in Seg n holds v.j = l.(A.j)
proof
  let A be FinSequence of n-tuples_on BOOLEAN,
  B be finite Subset of n-BinaryVectSp,
  v be Element of n-tuples_on BOOLEAN;
  assume that
A1: rng A = B and
A2: len A = n and
A3: A is one-to-one;
  set V = n-BinaryVectSp;
  defpred P1[object,object] means ex j be Nat st j in Seg n &
  $1 = A.j & $2 = v.j;
A4: for x being object st x in B ex y being object st
  y in the carrier of Z_2 & P1[x,y]
  proof
    let x be object;
    assume
A5: x in B;
    consider j be object such that
A6: j in dom A & x = A.j by A1,A5,FUNCT_1:def 3;
A7: j in Seg n by A2,A6,FINSEQ_1:def 3;
    reconsider j as Nat by A6;
    v.j in the carrier of Z_2 by BSPACE:3;
    hence thesis by A6,A7;
  end;
  consider l0 being Function of B,the carrier of Z_2 such that
A8: for x being object st x in B holds P1[x,l0.x] from FUNCT_2:sch 1(A4);
A9: for j be Nat st j in Seg n holds l0.(A.j) = v.j
  proof
    let j be Nat;
    assume j in Seg n;
    then
A10: j in dom A by A2,FINSEQ_1:def 3;
    then consider i be Nat such that
A11: i in Seg n & A.j = A.i & l0.(A.j) = v.i by A1,FUNCT_1:3,A8;
    i in dom A by A2,A11,FINSEQ_1:def 3;
    hence l0.(A.j) = v.j by A3,A10,A11,FUNCT_1:def 4;
  end;
  set f = (the carrier of V) --> 0.Z_2;
  set l = f +* l0;
A12: dom l = (dom f) \/ (dom l0) by FUNCT_4:def 1
  .= (the carrier of V) \/ (dom l0)
  .= (the carrier of V) \/ B by FUNCT_2:def 1
  .= (the carrier of V) by XBOOLE_1:12;
  rng l c= the carrier of Z_2;
  then l is Function of (the carrier of V),the carrier of Z_2
  by FUNCT_2:2,A12;
  then reconsider l as Element of Funcs (the carrier of V,
  the carrier of Z_2) by FUNCT_2:8;
A13: for v being Element of V st not v in B holds l.v = 0.Z_2
  proof
    let v be Element of V;
    v in dom l by A12;
    then
A14: v in (dom f) \/ (dom l0) by FUNCT_4:def 1;
    assume not v in B;
    then not v in dom l0;
    then l.v = f.v by A14,FUNCT_4:def 1;
    hence l.v = 0.Z_2;
  end;
  then reconsider l as Linear_Combination of V by VECTSP_6:def 1;
  for x be object st x in Carrier l holds x in B
  proof
    let x be object;
    assume x in Carrier l;
    then ex v be Element of V st x = v & l.v <> 0.Z_2;
    hence x in B by A13;
  end;
  then
A15: l is Linear_Combination of B by TARSKI:def 3,VECTSP_6:def 4;
  for j be Nat st j in Seg n holds v.j = l.(A.j)
  proof
    let j be Nat;
    assume
A16: j in Seg n;
    then j in dom A by A2,FINSEQ_1:def 3;
    then
A17: A.j in B by A1,FUNCT_1:3;
    then reconsider x = A.j as Element of V;
A18: x in dom l0 by FUNCT_2:def 1,A17;
    x in dom l by A12;
    then
A19: x in (dom f) \/ (dom l0) by FUNCT_4:def 1;
    thus l.(A.j) = l0.x by A18,A19,FUNCT_4:def 1
    .= v.j by A9,A16;
  end;
  hence thesis by A15;
end;
