
theorem Th12:
  for G being finite Group, p being Prime holds (
for H being Subgroup of G st H is p-group holds ex P being Subgroup
  of G st P is_Sylow_p-subgroup_of_prime p & H is Subgroup of P) & (for P1,P2
  being Subgroup of G st P1 is_Sylow_p-subgroup_of_prime p & P2
  is_Sylow_p-subgroup_of_prime p holds P1,P2 are_conjugated)
proof
  let G be finite Group;
  let p be Prime;
  reconsider p9=p as prime Element of NAT by ORDINAL1:def 12;
A1: for H,P being Subgroup of G st H is p-group & P
is_Sylow_p-subgroup_of_prime p holds ex g being Element of G st carr H c= carr(
  P |^ g)
  proof
    let H,P be Subgroup of G;
    set H9= H;
    set E = Left_Cosets P;
    set T = the_left_operation_of(H9,P);
    reconsider E as non empty finite set;
    reconsider T as LeftOperation of H9, E;
    assume H is p-group;
    then
A2: card the_fixed_points_of T mod p = card E mod p by Th9;
    assume P is_Sylow_p-subgroup_of_prime p;
    then
A3: not p divides index P;
    now
      assume card E mod p = 0;
      then (ex t being Nat st card E = p * t + 0 & 0<p) or 0=0 & p=0 by
NAT_D:def 2;
      then p9 divides card E by NAT_D:def 3;
      hence contradiction by A3,GROUP_2:def 18;
    end;
    then card the_fixed_points_of T <> 0 by A2,NAT_D:26;
    then consider x be object such that
A4: x in the_fixed_points_of T by CARD_1:27,XBOOLE_0:def 1;
    x in {x9 where x9 is Element of E: x9 is_fixed_under T} by A4,Def13;
    then consider x9 be Element of E such that
    x=x9 and
A5: x9 is_fixed_under T;
    x9 in Left_Cosets P;
    then consider g9 be Element of G such that
A6: x9 = g9 * P by GROUP_2:def 15;
    set g=g9";
    take g;
    now
      reconsider P1=x9 as Element of Left_Cosets P;
      let y be object;
      assume y in carr H;
      then reconsider h=y as Element of H9;
      reconsider h9=h as Element of H;
      consider P2 be Element of Left_Cosets P, A1,A2 be Subset of G, g99 be
      Element of G such that
A7:   P2 = the_left_translation_of(h9,P).P1 & A2 = g99 * A1 & A1 = P1
      & A2 = P2 and
A8:   g99 = h9 by Def8;
      the_left_operation_of(H,P).h9 = the_left_translation_of(h9,P) by Def9;
      then
A9:   g9 * P * g9" = g99 * (g9 * P) * g9" by A5,A6,A7
        .= g99 * (g9 * P * g9") by GROUP_2:33;
      g99 in g9 * P * g9"
      proof
        1_G in P by GROUP_2:46;
        then
A10:    1_G in carr P;
        g9 = g9 * 1_G by GROUP_1:def 4;
        then g9 * g9" = 1_G & g9 in g9 * P by A10,GROUP_1:def 5,GROUP_2:27;
        then
A11:    g99 = g99 * 1_G & 1_G in (g9 * P * g9") by GROUP_1:def 4,GROUP_2:28;
        assume not g99 in g9 * P * g9";
        hence contradiction by A9,A11,GROUP_2:27;
      end;
      hence y in g9 * P * g9" by A8;
    end;
    then carr H c= g" * P * g;
    hence thesis by GROUP_3:59;
  end;
  thus for H being Subgroup of G st H is p-group holds ex P being
  Subgroup of G st P is_Sylow_p-subgroup_of_prime p & H is Subgroup of P
  proof
    let H be Subgroup of G;
    consider P9 be strict Subgroup of G such that
A12: P9 is_Sylow_p-subgroup_of_prime p by Th10;
    assume H is p-group;
    then consider g be Element of G such that
A13: carr H c= carr(P9 |^ g) by A1,A12;
    set P = P9 |^ g;
    take P;
    set H2 = P;
    reconsider H2 as finite Group;
A14: card P9 = card P by GROUP_3:66;
    P9 is p-group by A12;
    then consider H1 be finite Group such that
A15: P9 = H1 and
A16: H1 is p-group;
    ex r be Nat st card H1 = p |^ r by A16;
    then
A17: H2 is p-group by A15,A14;
A18: not p divides index P9 by A12;
    card P * index P = card G by GROUP_2:147
      .= card P * index P9 by A14,GROUP_2:147;
    then not p divides index P by A18,XCMPLX_1:5;
    hence P is_Sylow_p-subgroup_of_prime p by A17;
    thus thesis by A13,GROUP_2:57;
  end;
  let P1,P2 be Subgroup of G;
  assume
A19: P1 is_Sylow_p-subgroup_of_prime p;
  then
A20: P1 is p-group;
  then consider H1 be finite Group such that
A21: P1 = H1 and
A22: H1 is p-group;
A23: not p divides index P1 by A19;
  consider r1 be Nat such that
A24: card H1 = p |^ r1 by A22;
  assume
A25: P2 is_Sylow_p-subgroup_of_prime p;
  then
A26: not p divides index P2;
  P2 is p-group by A25;
  then consider H2 be finite Group such that
A27: P2 = H2 and
A28: H2 is p-group;
  consider r2 be Nat such that
A29: card H2 = p |^ r2 by A28;
A30: card G = card P2 * index P2 by GROUP_2:147;
  then
A31: p|^r1 * index P1=p|^r2 * index P2 by A21,A24,A27,A29,GROUP_2:147;
  now
    assume
A32: r1<>r2;
    per cases by A32,XXREAL_0:1;
    suppose
A33:  r1 > r2;
      set r = r1 -' r2;
A34:  r1 - r2 > r2 - r2 by A33,XREAL_1:9;
      then
A35:  r = r1 - r2 by XREAL_0:def 2;
      then consider k be Nat such that
A36:  r = k + 1 by A34,NAT_1:6;
      r1 = r2 + r by A35;
      then p9 |^ r1 = p9 |^r2 * p9 |^ r by NEWTON:8;
      then p9 |^ r2 * (p9 |^ r * index P1) = p9 |^ r2 * index P2 by A31;
      then p9 |^ r * index P1 = index P2 by XCMPLX_1:5;
      then p |^ k * p * index P1 = index P2 by A36,NEWTON:6;
      then p * (p |^ k * index P1) = index P2;
      hence contradiction by A26,NAT_D:def 3;
    end;
    suppose
A37:  r1<r2;
      set r = r2 -' r1;
A38:  r2 - r1 > r1 - r1 by A37,XREAL_1:9;
      then
A39:  r = r2 - r1 by XREAL_0:def 2;
      then consider k be Nat such that
A40:  r = k + 1 by A38,NAT_1:6;
      r2 = r1 + r by A39;
      then p9 |^ r2 = p9 |^r1 * p9 |^ r by NEWTON:8;
      then p9 |^ r1 * (p9 |^ r * index P2) = p9 |^ r1 * index P1 by A21,A24,A27
,A29,A30,GROUP_2:147;
      then p9 |^ r * index P2 = index P1 by XCMPLX_1:5;
      then p |^ k * p * index P2 = index P1 by A40,NEWTON:6;
      then p * (p |^ k * index P2) = index P1;
      hence contradiction by A23,NAT_D:def 3;
    end;
  end;
  then
A41: card carr P1 = card carr P2 by A21,A24,A27,A29;
  consider g be Element of G such that
A42: carr P1 c= carr(P2 |^ g) by A1,A20,A25;
  card carr P2 = card P2 .= card(P2 |^ g) by GROUP_3:66
    .= card carr(P2 |^ g);
  then the multMagma of P1 = the multMagma of P2 |^ g
  by A42,A41,CARD_2:102,GROUP_2:59;
  hence thesis by GROUP_3:def 11;
end;
