
theorem Th14:
  for T be full Tree for n be non zero Nat for y be
  Element of n-tuples_on BOOLEAN st y = 0*n
  holds NumberOnLevel(n,T).'not' y = 2 to_power n
proof
  let T be full Tree;
  let n be non zero Nat;
  let y be Element of n-tuples_on BOOLEAN;
  assume
A1: y = 0*n;
  then
A2: Rev 'not' y = 'not' y by BINARI_3:9;
  len Rev 'not' y = len 'not' y by FINSEQ_5:def 3
    .= n by CARD_1:def 7;
  then reconsider F = Rev 'not' y as
  Element of n-tuples_on BOOLEAN by FINSEQ_2:92;
  reconsider ny = 'not' y as
  Element of n-tuples_on BOOLEAN by FINSEQ_2:131;
  T = {0,1}* by Def2;
  then ny in T-level n by Th11;
  hence NumberOnLevel(n,T).'not' y = (Absval F) + 1 by Def1
    .= 2 to_power n - 1 + 1 by A1,A2,BINARI_3:7
    .= 2 to_power n;
end;
