reserve G for Group,
  a,b for Element of G,
  m, n for Nat,
  p for Prime;

theorem
  for G being finite Group,N1,N2 being normal Subgroup of G holds
  N2 is Subgroup of center G & N1 is p-commutative-group &
  N2 is p-commutative-group implies
  ex N being strict normal Subgroup of G
  st the carrier of N = N1 * N2 & N is p-commutative-group
proof
  let G be finite Group;
  let N1,N2 be normal Subgroup of G;
  assume that
A1: N2 is Subgroup of center G and
A2: N1 is p-commutative-group & N2 is p-commutative-group;
  consider N be strict normal Subgroup of G such that
A3: the carrier of N = N1 * N2 & N is p-group by A2,Th21;
 for a,b being Element of N holds (a * b) |^ p = a |^ p * (b |^ p)
  proof
    let a,b be Element of N;
A4: a in N1 * N2 & b in N1 * N2 by A3;
    then consider a1,a2 be Element of G such that
A5: a = a1 * a2 & a1 in N1 & a2 in N2 by GROUP_11:6;
A6: a2 in center G by A1,A5,GROUP_2:40;
    consider b1,b2 be Element of G such that
A7: b = b1 * b2 & b1 in N1 & b2 in N2 by A4,GROUP_11:6;
A8: b2 in center G by A1,A7,GROUP_2:40;
    then a2 * b2 in center G & b2 * a2 in center G by A6,GROUP_2:50;
    then
A9: (a1 * b1) * (b2 * a2) = (b2 * a2) * (a1 * b1) by GROUP_5:77;
    reconsider c1 = a1,d1 = b1 as Element of N1 by A5,A7;
A10: a1 |^ p = c1 |^ p & b1 |^ p = d1 |^ p by GROUP_4:2;
    a1 * b1 = c1 * d1 by GROUP_2:43; then
A11: (a1 * b1) |^ p = (c1 * d1) |^ p by GROUP_8:4
                   .= c1 |^ p * (d1 |^ p) by A2,Def3
                   .= a1 |^ p * (b1 |^ p) by A10,GROUP_2:43;
    reconsider c2 = a2,d2 = b2 as Element of N2 by A5,A7;
A12: a2 |^ p = c2 |^ p & b2 |^ p = d2 |^ p by GROUP_4:2;
    b2 * a2 = d2 * c2 by GROUP_2:43; then
A13: (b2 * a2) |^ p = (d2 * c2) |^ p by GROUP_8:4
                   .= d2 |^ p * (c2 |^ p) by A2,Def3
                   .= b2 |^ p * (a2 |^ p) by A12,GROUP_2:43;
A14: a2 |^ p in center G by A6,GROUP_4:4;
A15: a1 * a2 = a2 * a1 by A6,GROUP_5:77;
A16: b1 * b2 = b2 * b1 by A8,GROUP_5:77;
    a * b = (a1 * a2) * (b1 * b2) by A5,A7,GROUP_2:43; then
A17: (a * b) |^ p = (a1 * a2 * (b1 * b2))|^ p by GROUP_8:4
             .= (a1 * (a2 * (b1 * b2)))|^ p by GROUP_1:def 3
             .= (a1 * ((b1 * b2) * a2))|^ p by A6,GROUP_5:77
             .= (a1 * (b1 * (b2 * a2)))|^ p by GROUP_1:def 3
             .= ((a1 * b1) * (b2 * a2))|^ p by GROUP_1:def 3
             .= (a1 * b1)|^ p * ((b2 * a2)|^ p) by A9,GROUP_1:38
             .= a1 |^ p * (b1 |^ p) * (a2 |^ p * (b2 |^ p)) by A11,A13,A14,
GROUP_5:77
             .= a1 |^ p * (b1 |^ p * (a2 |^ p * (b2 |^ p))) by GROUP_1:def 3
             .= a1 |^ p * ((b1 |^ p * (a2 |^ p)) * (b2 |^ p)) by GROUP_1:def 3
             .= a1 |^ p * ((a2 |^ p * (b1 |^ p)) * (b2 |^ p)) by A14,GROUP_5:77
             .= (a1 |^ p * (a2 |^ p * (b1 |^ p))) * (b2 |^ p) by GROUP_1:def 3
             .= (a1 |^ p * (a2 |^ p) * (b1 |^ p)) * (b2 |^ p) by GROUP_1:def 3
             .= (a1 |^ p * (a2 |^ p)) * (b1 |^ p * (b2 |^ p)) by GROUP_1:def 3
             .= (a1 * a2)|^ p * (b1 |^ p * (b2 |^ p)) by A15,GROUP_1:38
             .= (a1 * a2)|^ p * ((b1 * b2) |^ p) by A16,GROUP_1:38;
A18:   a |^ p = (a1 * a2)|^ p by A5,GROUP_4:2;
       b |^ p = (b1 * b2)|^ p by A7,GROUP_4:2;
    hence thesis by A17,A18,GROUP_2:43;
  end;
  then N is p-commutative-group-like;
  hence thesis by A3;
end;
