
theorem Th16:
  for G being finite commutative Group,
  h,k be Nat
  st card G = h*k & h,k are_coprime holds
  ex H,K being strict finite Subgroup of G st
  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}
  proof
    let G be finite commutative Group,
    h,k be Nat;
    assume A1:card G = h*k & h,k are_coprime;
    set A = {x where x is Element of G: x|^h = 1_G};
    set B = {x where x is Element of G: x|^k = 1_G};
    A c= the carrier of G
    proof
      let y be object;
      assume y in A; then
      ex x be Element of G st y=x & x|^h = 1_G;
      hence y in the carrier of G;
    end; then
    reconsider A as Subset of G;
    B c= the carrier of G
    proof
      let y be object;
      assume y in B; then
      ex x be Element of G st y=x & x|^k = 1_G;
      hence y in the carrier of G;
    end; then
    reconsider B as Subset of G;
    consider H being strict finite Subgroup of G such that A2:
    the carrier of H = A & H is commutative
    & H is normal by Th14;
    consider K being strict finite Subgroup of G
    such that A3:
    the carrier of K = B & K is commutative & K is normal by Th14;
    (1_G) |^ h = 1_G by GROUP_1:31;
    then
    A4: 1_G in the carrier of H by A2;
    (1_G) |^ k = 1_G by GROUP_1:31;
    then
    1_G in the carrier of K by A3;
    then
    1_G in (the carrier of H) /\ (the carrier of K)
    by A4,XBOOLE_0:def 4;
    then
    A5: {1_G} c= (the carrier of H) /\ (the carrier of K)
    by ZFMISC_1:31;
    h gcd k = 1 by A1,INT_2:def 3;
    then
    consider a,b be Integer such that
    A6: a*h + b*k = 1 by NAT_D:68;
    (the carrier of H) /\ (the carrier of K) c= {1_G}
    proof
      let z be object;
      assume A7: z in (the carrier of H) /\ (the carrier of K);
      then
      z in the carrier of H by XBOOLE_0:def 4;
      then
      z in G by STRUCT_0:def 5,GROUP_2:40;
      then
      reconsider w=z as Element of G;
      A8: w in A & w in B by A2,A3,A7,XBOOLE_0:def 4;
      then
      A9: ex x be Element of G st w=x & x|^h = 1_G;
      A10: ex x be Element of G st w=x & x|^k = 1_G by A8;
      w = w|^1 by GROUP_1:26
      .= (w|^(a*h)) * (w|^(b*k)) by GROUP_1:33,A6
      .= ((w|^h) |^a) *(w|^(b*k)) by GROUP_1:35
      .= ((w|^h) |^a) * ((w|^k) |^b) by GROUP_1:35
      .= 1_G * ((1_G) |^b) by GROUP_1:31,A10,A9
      .= 1_G * 1_G by GROUP_1:31
      .= 1_G by GROUP_1:def 4;
      hence z in {1_G} by TARSKI:def 1;
    end;
    then
    A11: (the carrier of H) /\ (the carrier of K) c= {1_G};
    A12: for x be Element of G holds
    ex s,t be Element of G
    st s in H & t in K & x = s*t
    proof
      let x be Element of G;
      A13: x = x|^1 by GROUP_1:26
      .=(x|^(b*k)) * (x|^(a*h)) by GROUP_1:33,A6;
      set t = x|^(a*h);
      set s = x|^(b*k);
      s |^h = x|^(b*k*h) by GROUP_1:35
      .= x|^((k*h)*b)
      .=(x|^(k*h)) |^ b by GROUP_1:35
      .= (1_G) |^ b by A1,GR_CY_1:9
      .= 1_G by GROUP_1:31;
      then
      A14: s in H by A2;
      t |^k = x|^(a*h*k) by GROUP_1:35
      .= x|^((h*k)*a)
      .=(x|^(h*k)) |^ a by GROUP_1:35
      .= (1_G) |^ a by A1,GR_CY_1:9
      .= 1_G by GROUP_1:31;
      then t in K by A3;
      hence thesis by A13,A14;
    end;
    take H,K;
    thus thesis by A2,A3,A11,A12,A5,XBOOLE_0:def 10;
  end;
