reserve n for non zero Nat,
  j,k,l,m for Nat,
  g,h,i for Integer;

theorem Th12:
  for n be non zero Nat holds for z be Tuple of n,BOOLEAN st z/.n
  = TRUE holds Absval(z) >= 2 to_power (n-'1)
proof
  defpred P[Nat] means (for z be Tuple of $1,BOOLEAN st z/.$1 = TRUE holds
  Absval(z) >= 2 to_power ($1-'1));
A1: for n be non zero Nat st P[n] holds P[n+1]
  proof
    let n be non zero Nat such that
    P[n];
    let z be Tuple of (n+1),BOOLEAN such that
A2: z/.(n+1) = TRUE;
    consider x be Element of n-tuples_on BOOLEAN, a be Element of BOOLEAN
    such that
A3: z = x^<*a*> by FINSEQ_2:117;
A4: n + 1 >= 1 by NAT_1:11;
    then n + 1 - 1 >= 1 - 1 by XREAL_1:9;
    then
A5: n + 1 -' 1 = n by XREAL_0:def 2;
    len z = n+1 by CARD_1:def 7;
    then
A6: z/.(n+1) = (x^<*a*>).(n+1) by A3,A4,FINSEQ_4:15
      .= a by FINSEQ_2:116;
    Absval(z) = Absval(x) + IFEQ(a,FALSE,0,2 to_power n) by A3,BINARITH:20
      .= Absval(x) + 2 to_power n by A2,A6,FUNCOP_1:def 8;
    hence thesis by A5,NAT_1:11;
  end;
A7: P[1]
  proof
    let z be Tuple of 1,BOOLEAN such that
A8: z/.1 = TRUE;
A9: len z = 1 by CARD_1:def 7;
    then z.1 = z/.1 by FINSEQ_4:15;
    then z = <*TRUE*> by A8,A9,FINSEQ_1:40;
    then
A10: Absval(z) = 1 by BINARITH:16;
    2 to_power (1-'1) = 2 to_power (1-1) by XREAL_0:def 2;
    hence thesis by A10,POWER:24;
  end;
  thus for n being non zero Nat holds P[n] from NAT_1:sch 10(A7,A1);
end;
