 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
reserve G1,G2 for Group;

theorem :: TH123
  for n being even non zero Nat st n > 2 holds
    INT.Group 2, center Dihedral_group n are_isomorphic
proof
  let n be even non zero Nat;
  assume A1: n > 2;
  consider k being Nat such that
  A2: n = 2*k by ABIAN:def 2;
  2 < k*2 by A1, A2;
  then 2*(1/2) < (2*k)*(1/2) by XREAL_1:68;
  then A3: 1 < k;
  1 < 2 & 2 < n by A1;
  then 1 < n by XXREAL_0:2;
  then 1 in Segm n by NAT_1:44;
  then 1 in INT.Group n by Th76;
  then reconsider g1 = 1 as Element of INT.Group n;
  reconsider x = <* g1, 1_(INT.Group 2) *> as Element of Dihedral_group n
    by Th9;
  for z being object
  holds z in the carrier of center Dihedral_group n
  iff z in {1_(Dihedral_group n), x |^ k}
  proof
    let z be object;
    B1: g1 = 1 & x = <* g1, 1_(INT.Group 2) *>;
    hereby
      assume z in the carrier of center Dihedral_group n;
      then B2: z in center Dihedral_group n;
      then z in Dihedral_group n by GROUP_2:40;
      then reconsider z1=z as Element of Dihedral_group n;
      z1 in center Dihedral_group n by B2;
      then z1 = 1_(Dihedral_group n) or z1 = x |^ k by A1, A2, Th122;
      hence z in {1_(Dihedral_group n), x |^ k} by TARSKI:def 2;
    end;
    assume B3: z in {1_(Dihedral_group n), x |^ k};
    then z = 1_(Dihedral_group n) or z = x |^ k by TARSKI:def 2;
    then reconsider z1=z as Element of Dihedral_group n;
    z1 = 1_(Dihedral_group n) or z1 = x |^ k by B3, TARSKI:def 2;
    then z1 in center Dihedral_group n by A1, A2, B1, Th122;
    hence z in the carrier of center Dihedral_group n;
  end;
  then the carrier of center Dihedral_group n c= {1_(Dihedral_group n), x |^ k}
  & {1_(Dihedral_group n), x |^ k} c= the carrier of center Dihedral_group n;

  then
  A4: the carrier of center Dihedral_group n = {1_(Dihedral_group n), x |^ k}
  by XBOOLE_0:def 10;
  A5: card (center Dihedral_group n) = 2
  proof
    0 < k by A3;
    then k < k + k by XREAL_1:29;
    then k < n & k <> 0 by A2;
    then x |^ k <> 1_(Dihedral_group n) by Th102;
    then card {1_(Dihedral_group n), x |^ k} = 2 by CARD_2:57;
    hence thesis by A4;
  end;
  card (INT.Group 2) = 2;
  hence INT.Group 2, center Dihedral_group n are_isomorphic by A5, GR_CY_2:19;
end;
