
theorem
  for n,m be Element of NAT st 0 < n holds
  ALGO_BPOW(0,n,m) = 0
  proof
    let n,m be Element of NAT;
    assume AS: 0 < n;
    consider A,B be sequence of NAT such that
    ASC:
    ALGO_BPOW(0,n,m) = B. (LenBSeq n) &
    A.0 = 0 mod m & B.0 = 1 &
    ( for i be Nat holds
    A.(i+1) = (A.i)*(A.i) mod m &
    B.(i+1) = BinBranch((B.i),(B.i)*(A.i) mod m,(Nat2BL.n).(i+1))) by Def1;
    (LenBSeq n)-1 in NAT by INT_1:5,NAT_1:14;
    then reconsider fs = (LenBSeq n)-1 as Nat;
    QW:A.fs = 0 to_power (2 to_power fs) mod m by CBPOW1,ASC
    .= 0 mod m by POWER:42;
    ALGO_BPOW(0,n,m)
     = BinBranch((B.fs),(B.fs)*(A.fs) mod m,(Nat2BL.n).(fs+1)) by ASC
    .= BinBranch((B.fs),(B.fs)*(A.fs) mod m,1) by MMS1,AS
    .= 0 by QW,defBB;
    hence thesis;
  end;
