
theorem Th12:
  for G being finite Group, a being Element of G
  holds card the set of all  a-con_map"{x} where x is Element of con_class a
 = card con_class 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;
  deffunc FF(object) = a-con_map"{$1};
A1: for x being object st x in con_class a holds FF(x) in X;
  consider F being Function of con_class a, X such that
A2: for x being object st x in con_class a holds F.x = FF(x)
from FUNCT_2:sch 2
  (A1);

  for c being object st c in X
ex x being object st x in con_class a & c = F.x
  proof
    let c be object such that
A3: c in X;
    reconsider c as Subset of G by A3;
    consider y being Element of con_class a such that
A4: c = a-con_map"{y} by A3;
    F.y = c by A2,A4;
    hence thesis;
  end;
  then
A5: rng F = X by FUNCT_2:10;
A6: dom F = con_class a by FUNCT_2:def 1;
  for x1,x2 being object
st x1 in dom F & x2 in dom F & F.x1=F.x2 holds x1 = x2
  proof
    let x1,x2 be object such that
A7: x1 in dom F and
A8: x2 in dom F and
A9: F.x1 = F.x2;
    reconsider y1=x1 as Element of con_class a by A7;
    reconsider y2=x2 as Element of con_class a by A8;
A10: a-con_map"{y1} = F.y1 by A2;
A11: a-con_map"{y2} = F.y2 by A2;
    now
      assume y1<>y2;
      then (a-con_map"{y1}) misses (a-con_map"{y2}) by Th10;
      then ((a-con_map"{y1}) /\ (a-con_map"{y2})) = {} by XBOOLE_0:def 7;
      hence contradiction by A9,A10,A11;
    end;
    hence thesis;
  end;
  then F is one-to-one;
  then con_class a, F.:(con_class a) are_equipotent by A6,CARD_1:33;
  then con_class a, rng F are_equipotent by A6,RELAT_1:113;
  hence thesis by A5,CARD_1:5;
end;
