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 Th15:
  for z1 being Tuple of 1,BOOLEAN holds z1=<*FALSE*> implies Absval(z1) = 0
proof
  let z1 be Tuple of 1,BOOLEAN;
A1: ex k being Element of NAT st Binary( z1 ) = <* k *> by FINSEQ_2:97;
  assume z1=<*FALSE*>;
  then
A2: z1/.1 = FALSE by FINSEQ_4:16;
  1 in Seg 1;
  then (Binary(z1))/.1 = IFEQ(z1/.1,FALSE,0,2 to_power(1-'1)) by Def3
    .= 0 by A2,FUNCOP_1:def 8;
  hence Absval(z1) = addnat $$ <* 0 *> by A1,FINSEQ_4:16
    .= 0 by FINSOP_1:11;
end;
