reserve i,n for Nat;
reserve m for non zero Nat;
reserve p,q for Tuple of n, BOOLEAN;
reserve d,d1,d2 for Element of BOOLEAN;

theorem Th8:
  for m holds Intval (Bin1(m)^<*FALSE*>) = 1
proof
  defpred P[non zero Nat] means Intval (Bin1($1)^<*FALSE*>) = 1;
A1: P[1]
  proof
    consider k being Element of BOOLEAN such that
A2: Bin1(1) = <* k *> by FINSEQ_2:97;
A3: (Bin1(1))/.1 = k by A2,FINSEQ_4:16;
 1 in Seg 1 by FINSEQ_1:3;
then  Bin1(1) = <*TRUE*> by A2,A3,Th5;
    hence thesis by Th2;
  end;
A4: now
    let m be non zero Nat such that
A5: P[m];
 (Bin1(m)^<*FALSE*>)/.(m+1) = FALSE by BINARITH:2;
then A6: Absval(Bin1(m)^<*FALSE*>) = 1 by A5,Def3;
 (Bin1(m+1)^<*FALSE*>)/.(m+1+1) = FALSE by BINARITH:2;
then  Intval(Bin1(m+1)^<*FALSE*>) = Absval(Bin1(m+1)^<*FALSE*>) by Def3
      .= Absval(Bin1(m)^<*FALSE*>^<*FALSE*>) by Th7
      .= Absval(Bin1(m)^<*FALSE*>)
    + IFEQ(FALSE,FALSE,0,2 to_power(m+1)) by BINARITH:20
      .= 1 + 0 by A6,FUNCOP_1:def 8
      .= 1;
    hence P[m+1];
  end;
  thus for m holds P[m] from NAT_1:sch 10(A1,A4);
end;
