
theorem Th13:
  for T be full Tree for n be non zero Nat holds
  NumberOnLevel(n,T).(0*n) = 1
proof
  let T be full Tree;
  let n be non zero Nat;
A1: len Rev (0*n) = len 0*n by FINSEQ_5:def 3
    .= n by CARD_1:def 7;
A2: T = {0,1}* by Def2;
  then 0*n is Element of T by BINARI_3:4;
  then Rev (0*n) is FinSequence of BOOLEAN by Lm3;
  then reconsider F = Rev (0*n) as
   Element of n-tuples_on BOOLEAN by A1,FINSEQ_2:92;
  0*n in T-level n by A2,Th10;
  hence NumberOnLevel(n,T).(0*n) = (Absval F) + 1 by Def1
    .= 0 + 1 by BINARI_3:6,8
    .= 1;
end;
