:: deftheorem Def4 defines ALGO_CRT NTALGO_1:def 4 :
for nlist being non empty FinSequence of [:INT,INT:]
for b₂ being Element of INT holds
( b₂ = ALGO_CRT nlist iff ( ( len nlist = 1 implies b₂ = (nlist . 1) `1 ) & ( len nlist <> 1 implies ex m, n, prodc, prodi being FinSequence of INT ex M0, M being Element of INT st
( len m = len nlist & len n = len nlist & len prodc = (len nlist) - 1 & len prodi = (len nlist) - 1 & m . 1 = 1 & ( for i being Nat st 1 <= i & i <= (len m) - 1 holds
ex d, x, y being Element of INT st
( x = (nlist . i) `2 & m . (i + 1) = (m . i) * x & y = m . (i + 1) & d = (nlist . (i + 1)) `2 & prodi . i = ALGO_INVERSE (y,d) & prodc . i = y ) ) & M0 = (nlist . (len m)) `2 & M = (prodc . ((len m) - 1)) * M0 & n . 1 = (nlist . 1) `1 & ( for i being Nat st 1 <= i & i <= (len m) - 1 holds
ex u, u0, u1 being Element of INT st
( u0 = (nlist . (i + 1)) `1 & u1 = (nlist . (i + 1)) `2 & u = ((u0 - (n . i)) * (prodi . i)) mod u1 & n . (i + 1) = (n . i) + (u * (prodc . i)) ) ) & b₂ = (n . (len m)) mod M ) ) ) );