
theorem Th9: :: Br1:
for G being finite Group, a being Element of G, x being Element of con_class a
  holds card (a-con_map"{x}) = card Centralizer a
proof
  let G be finite Group, a be Element of G, x be Element of con_class a;
  ex b being Element of G st ( b = x)&( a,b are_conjugated) by GROUP_3:80;
  then consider d being Element of G such that
A1: x = a |^ d by GROUP_3:74;
  reconsider Cad = (Centralizer a)*d as Subset of G;
A2: ex B,C being finite set st ( B = d*(Centralizer a))&( C =
  Cad)&( card Centralizer a = card B)&( card Centralizer a = card C) by
GROUP_2:133;
  for g being object holds g in a-con_map"{x} iff g in Cad
  proof
    let g be object;
A3: now
      assume
A4:   g in a-con_map"{x};
      then a-con_map.g in {x} by FUNCT_1:def 7;
      then
A5:   a-con_map.g = x by TARSKI:def 1;
      reconsider y=g as Element of G by A4;
A6:   a-con_map.g = a |^ y by Def2;
A7:   y*((d"*a)*d) = y*(d"*a)*d by GROUP_1:def 3
        .= y*d"*a*d by GROUP_1:def 3;
      y*((y"*a)*y) = y*(y"*a)*y by GROUP_1:def 3
        .= a*y by GROUP_3:1;
      then y*d"*a*d*d" = a*(y*d") by A1,A5,A6,A7,GROUP_1:def 3;
      then (y*d")*a = a*(y*d") by GROUP_3:1;
      then (y*d") is Element of Centralizer a by Th8;
      then consider z being Element of G such that
A8:   z in the carrier of Centralizer a and
A9:   y*d" = z;
A10:  z in Centralizer a by A8;
      reconsider z as Element of G;
      y = z*d by A9,GROUP_3:1;
      hence g in Cad by A10,GROUP_2:104;
    end;
    now
      assume g in Cad;
      then consider z being Element of G such that
A11:  g = z*d and
A12:  z in Centralizer a by GROUP_2:104;
      reconsider y=g as Element of G by A11;
      y*d" = z by A11,GROUP_3:1;
      then y*d" in carr(Centralizer a) by A12;
      then (y*d")*a = a*(y*d") by Th8;
      then (y*d")*a*d = a*((y*d")*d) by GROUP_1:def 3;
      then (y*d")*a*d = a*y by GROUP_3:1;
      then (y*d")*(a*d) = a*y by GROUP_1:def 3;
      then y"*((y*d")*(a*d)) = y"*a*y by GROUP_1:def 3;
      then (y"*(y*d"))*(a*d) = y"*a*y by GROUP_1:def 3;
      then d"*(a*d) = y"*a*y by GROUP_3:1;
      then x = a|^y by A1,GROUP_1:def 3;
      then a-con_map.y = x by Def2;
      then
A13:  a-con_map.y in {x} by TARSKI:def 1;
      dom (a-con_map) = the carrier of G by FUNCT_2:def 1;
      hence g in a-con_map"{x} by A13,FUNCT_1:def 7;
    end;
    hence thesis by A3;
  end;
  hence thesis by A2,TARSKI:2;
end;
