
theorem Th18:
  for G being finite commutative Group,
  h,k be non zero Nat
  st card G = h*k & h,k are_coprime
  ex H,K being strict finite Subgroup of G st
  card H = h & card K = k &
  (the carrier of H) /\ (the carrier of K) = {1_G} &
  ex F being Homomorphism of product <*H,K*>,G
  st F is bijective
  & for a,b be Element of G st a in H & b in K
  holds F.(<*a,b*>) = a*b
  proof
    let G be finite commutative Group,
    h,k be non zero Nat;
    assume A1: card G = h*k & h,k are_coprime;
    then
    consider H,K being strict finite Subgroup of G such that
    A2:
    the carrier of H = {x where x is Element of G: x|^h = 1_G} &
    the carrier of K = {x where x is Element of G: x|^k = 1_G} &
    H is normal & K is normal &
    (for x be Element of G holds
    ex a,b be Element of G st a in H & b in K & x = a*b) &
    (the carrier of H) /\ (the carrier of K) = {1_G} by Th16;
    take H,K;
    consider F being Homomorphism of product <*H,K*>,G such that
    A3: F is bijective & for a,b be Element of G st a in H & b in K
    holds F.(<*a,b*>) = a*b by A2,Th12;
    set s = card H;
    set t = card K;
    F is one-to-one & dom F = the carrier of product <*H,K*>
    & rng F = the carrier of G by A3,FUNCT_2:def 1,FUNCT_2:def 3; then
    card (product <*H,K*>) = card G by CARD_1:5,WELLORD2:def 4; then
    A4: s * t = h * k by A1,Th17;
    A5:for q being Prime st q in support (prime_factorization s)
    holds not q,h are_coprime by A2,Th15;
    for q being Prime st q in support (prime_factorization t)
    holds not q,k are_coprime by A2,Th15;
    hence thesis by A2,A3,A4,Th7,A5,A1;
  end;
