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

theorem Th6:
  for L be the carrier of (n-BinaryVectSp)-valued FinSequence,
  j be Nat
  st len L = m & m <= n & j in Seg n
  ex x be FinSequence of Z_2 st len x = m &
  for i be Nat st i in Seg m ex K be Element of n-tuples_on BOOLEAN
  st K = L.i & x.i = K.j
proof
  let L be the carrier of (n-BinaryVectSp)-valued FinSequence,
  j be Nat;
  assume
A1: len L = m & m <= n & j in Seg n;
  defpred P1[Nat,set] means ex K be Element of n-tuples_on BOOLEAN
  st K = L.$1 & $2 = K.j;
A2: for i be Nat st i in Seg m ex y being Element of BOOLEAN st P1[i,y]
  proof
    let i be Nat;
    assume i in Seg m;
    then i in dom L by A1,FINSEQ_1:def 3;
    then L/.i = L.i by PARTFUN1:def 6;
    then reconsider K = L.i as Element of n-tuples_on BOOLEAN;
    take K.j;
    thus P1[i,K.j];
  end;
  consider x being FinSequence of BOOLEAN such that
A3: dom x = Seg m & for i be Nat st i in Seg m holds P1[i,x.i]
  from FINSEQ_1:sch 5 (A2);
  len x = m by FINSEQ_1:def 3, A3;
  hence thesis by A3,BSPACE:3;
end;
