
theorem Th13:
  for G being finite Group, a being Element of G
  holds card G = card con_class a * card Centralizer a
proof
  let G be finite Group, a be Element of G;
  reconsider
  X = the set of all a-con_map"{x} where x is Element of con_class a
  as a_partition of the carrier of G by Th11;
  A1: for
 A being set st A in X holds A is finite & card A = card Centralizer a &
  for B being set st B in X & A<>B holds A misses B
  proof
    let A be set such that
A2: A in X;
    reconsider A as Subset of G by A2;
    ex x being Element of con_class a st ( A = a-con_map"{x}) by A2;
    hence thesis by A2,Th9,EQREL_1:def 4;
  end;
  reconsider k = card Centralizer a as Element of NAT;
  for Y being set st Y in X ex B being finite set st B = Y & card B = k &
  for Z being set st Z in X & Y <> Z holds Y,Z are_equipotent & Y misses Z
  proof
    let Y be set such that
A3: Y in X;
    reconsider Y as finite set by A3;
A4: card Y = card Centralizer a by A1,A3;
    for Z being set st Z in X & Y<>Z holds Y,Z are_equipotent & Y misses Z
    proof
      let Z be set such that
A5:   Z in X and
A6:   Y<>Z;
A7:   card Y = card Centralizer a by A1,A3;
      card Z = card Centralizer a by A1,A5;
      hence thesis by A1,A3,A5,A6,A7,CARD_1:5;
    end;
    hence thesis by A4;
  end;
  then consider C being finite set such that
A8: C = union X and
A9: card C = card X * k by GROUP_2:156;
  card G = card C by A8,EQREL_1:def 4
    .= card con_class a * card
  Centralizer a by A9,Th12;
  hence thesis;
end;
