
theorem Th28:
  for i being Nat holds (i+1)-BinarySequence 2 to_power i= 0*i^<*1 *>
proof
  deffunc Bi(Nat) = ($1+1)-BinarySequence 2 to_power $1;
  let i be Nat;
  set Bi = Bi(i);
  per cases;
  suppose
A1: i = 0;
    then
A2: 0*i = 0;
    reconsider i1 = i+1 as non zero Nat;
A3: 0*i1 = <*FALSE*> by A1,FINSEQ_2:59;
    then reconsider x = 0*i1 as Tuple of i1, BOOLEAN;
    2 to_power i1 = 2 by A1,POWER:25;
    then 1 = (2 to_power i1) - 1;
    then i1-BinarySequence 1 = 'not' x by Lm4;
    hence Bi = <*TRUE*> by A1,A3,Th14,POWER:24
      .= 0*i^<*1*> by A2,FINSEQ_1:34;
  end;
  suppose
    i > 0;
    then reconsider i9 = i as non zero Nat;
    Bi = 0*(i9)^<*1*> by Lm1;
    hence thesis;
  end;
end;
