
theorem Th11: :: Br3:
  for G being Group, a being Element of G
  holds the set of all  a-con_map"{x} where x is Element of con_class a
  is a_partition of the carrier of G
proof
  let G be 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 set;
A1: for y being object holds y in union X implies y in the carrier of G
  proof
    let x be object;
    assume x in union X;
    then consider Y being set such that
A2: x in Y and
A3: Y in X by TARSKI:def 4;
    ex z being Element of con_class a st ( a-con_map"{z} = Y) by A3;
    hence thesis by A2;
  end;
  for y being object holds y in the carrier of G implies y in union X
  proof
    let x be object;
    assume x in the carrier of G;
    then reconsider y=x as Element of G;
    consider z being Element of G such that
A4: z in con_class a and
A5: z = a|^y by GROUP_3:82;
    a-con_map.y = z by A5,Def2;
    then
A6: a-con_map.y in {z} by TARSKI:def 1;
    dom(a-con_map) = the carrier of G by FUNCT_2:def 1;
    then
A7: y in a-con_map"{z} by A6,FUNCT_1:def 7;
    a-con_map"{z} in X by A4;
    hence thesis by A7,TARSKI:def 4;
  end;
  then
A8: union X = the carrier of G by A1,TARSKI:2;
A9: for A being Subset of G st A in X holds A<>{} &
  for B being Subset of G st B in X holds A=B or A misses B
  proof
    let A be Subset of G;
    assume A in X;
    then consider x being Element of con_class a such that
A10: A = a-con_map"{x};
    a,x are_conjugated by GROUP_3:81;
    then consider g being Element of G such that
A11: x = a |^ g by GROUP_3:74;
    a-con_map.g = x by A11,Def2;
    then
A12: a-con_map.g in {x} by TARSKI:def 1;
A13: dom (a-con_map) = the carrier of G by FUNCT_2:def 1;
    for B being Subset of G st B in X holds A=B or A misses B
    proof
      let B be Subset of G;
      assume B in X;
      then ex y being Element of con_class a st ( B = a-con_map"{y});
      hence thesis by A10,Th10;
    end;
    hence thesis by A10,A12,A13,FUNCT_1:def 7;
  end;
  X c= bool union X by ZFMISC_1:82;
  hence thesis by A8,A9,EQREL_1:def 4;
end;
