reserve x,y,z for boolean object;
reserve i,j,k for Nat;
reserve n for non zero Nat;
reserve x,y,z1,z2 for Tuple of n, BOOLEAN;

theorem Th16:
  for z1 being Tuple of 1,BOOLEAN st z1=<*TRUE*> holds Absval(z1)= 1
proof
  let z1 be Tuple of 1,BOOLEAN;
A1: 1 - 1 = 0;
  assume z1=<*TRUE*>;
  then
A2: z1/.1 <> FALSE by FINSEQ_4:16;
  1 in Seg 1;
  then
A3: (Binary(z1))/.1 = IFEQ(z1/.1,FALSE,0,2 to_power(1-'1)) by Def3
    .= 2 to_power(1-'1) by A2,FUNCOP_1:def 8;
  ex k being Element of NAT st Binary( z1 ) = <* k *> by FINSEQ_2:97;
  hence Absval(z1) = addnat $$ <* 2 to_power(1-'1) *> by A3,FINSEQ_4:16
    .= 2 to_power(1-'1) by FINSOP_1:11
    .= 2 to_power(0) by A1,XREAL_0:def 2
    .= 1 by POWER:24;
end;
