let n, m be Element of NAT ; :: thesis: ( 0 < n implies ALGO_BPOW (0,n,m) = 0 )
assume AS: 0 < n ; :: thesis: ALGO_BPOW (0,n,m) = 0
consider A, B being sequence of NAT such that
ASC: ( ALGO_BPOW (0,n,m) = B . (LenBSeq n) & A . 0 = 0 mod m & B . 0 = 1 & ( for i being 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 ALGO_BPOW (0,n,m) = 0 ; :: thesis: verum