
theorem
  for G being finite Group, p being Prime holds card
  the_sylow_p-subgroups_of_prime(p,G) mod p = 1 & card
  the_sylow_p-subgroups_of_prime(p,G) divides card G
proof
  let G be finite Group;
  let p be Prime;
  set E = the_sylow_p-subgroups_of_prime(p,G);
A1: p > 1 by NAT_4:12;
  consider P be strict Subgroup of G such that
A2: P is_Sylow_p-subgroup_of_prime p by Th10;
  set P9 = (Omega).P;
  reconsider P9 as strict Subgroup of G;
  reconsider H9=P9 as Element of Subgroups G by GROUP_3:def 1;
  reconsider P9 as strict finite Subgroup of G;
  H9=P9;
  then P9 in E by A2;
  then reconsider P9 as Element of E;
  set T = the_left_operation_of(P,p);
A3: P is p-group by A2;
  set G9 = (Omega).G;
  set T9 = the_left_operation_of(G9,p);
  set K = the_strict_stabilizer_of(P9,T9);
A4: now
    reconsider P1=P9 as Element of E;
    reconsider P99=P9 as strict Subgroup of G;
    let y be object;
    thus y in the_orbit_of(P9,T9) implies y in E;
    assume y in E;
    then consider Q be Element of Subgroups G such that
A5: y=Q and
A6: ex P being strict Subgroup of G st P = Q & P
    is_Sylow_p-subgroup_of_prime p;
    consider Q9 be strict Subgroup of G such that
A7: Q9 = Q and
A8: Q9 is_Sylow_p-subgroup_of_prime p by A6;
    P99,Q9 are_conjugated by A2,A8,Th12;
    then consider g be Element of G such that
A9: Q9 = P99 |^ g by GROUP_3:def 11;
    Q9 in E by A7,A8;
    then reconsider Q99=Q9 as Element of E;
    reconsider g9=g" as Element of G9;
A10: g9" = (g")" by GROUP_2:48
      .= g;
    ex P2 be Element of E, H1,H2 be strict Subgroup of G, g999 be Element
of G st P2=the_left_translation_of(g9,p).P1 & P1=H1 & P2=H2 & g9"=g999 & H2 =
    H1 |^ g999 by Def20;
    then Q9 = (T9^g9).P9 by A9,A10,Def21;
    then P9,Q99 are_conjugated_under T9;
    hence y in the_orbit_of(P9,T9) by A5,A7;
  end;
  reconsider P as finite Subgroup of G;
  reconsider T as LeftOperation of P, E;
  for x being object holds x in the_fixed_points_of T iff x=P9
  proof
    let x be object;
    for h being Element of P holds P9 = (T^h).P9
    proof
      reconsider P1=P9 as Element of E;
      let h be Element of P;
      consider P2 be Element of E, H1,H2 be strict Subgroup of G, g be Element
      of G such that
A11:  P2=the_left_translation_of(h,p).P1 and
A12:  P1=H1 and
A13:  P2=H2 and
A14:  h"=g and
A15:  H2 = H1 |^ g by Def20;
A16:  g in H1 by A12,A14;
      now
        let y be object;
        assume
A17:    y in carr H1;
        then reconsider h9=y as Element of G;
A18:    h9 in H1 by A17;
        ex h being Element of G st y = h * g & h in g" * H1
        proof
          set h99 = h9*g";
          take h99;
          thus h99 * g = h9 * (g" * g) by GROUP_1:def 3
            .= h9 * 1_G by GROUP_1:def 5
            .= y by GROUP_1:def 4;
          g" in H1 by A16,GROUP_2:51;
          then
A19:      h99 in H1 by A18,GROUP_2:50;
          ex h being Element of G st h99 = g" * h & h in H1
          proof
            set h999 = g*h99;
            take h999;
            thus g" * h999 = g" * g * h99 by GROUP_1:def 3
              .= 1_G * h99 by GROUP_1:def 5
              .= h99 by GROUP_1:def 4;
            thus thesis by A16,A19,GROUP_2:50;
          end;
          hence thesis by GROUP_2:103;
        end;
        hence y in g" * H1 * g by GROUP_2:28;
      end;
      then carr H1 c= g" * H1 * g;
      then
A20:  carr H1 c= carr (H1 |^ g) by GROUP_3:59;
A21:  card carr H1 = card H1 .= card(H1 |^ g) by GROUP_3:66
        .= card carr(H1 |^ g);
      (T^h).P9=P2 by A11,Def21;
      hence thesis by A12,A13,A15,A20,A21,CARD_2:102,GROUP_2:59;
    end;
    then
A22: P9 is_fixed_under T;
    hereby
      assume x in the_fixed_points_of T;
      then x in {x9 where x9 is Element of E: x9 is_fixed_under T} by Def13;
      then consider Q be Element of E such that
A23:  x=Q and
A24:  Q is_fixed_under T;
      Q in {H99 where H99 is Element of Subgroups G: ex P being strict
      Subgroup of G st P = H99 & P is_Sylow_p-subgroup_of_prime p};
      then consider H99 be Element of Subgroups G such that
A25:  Q=H99 and
A26:  ex P being strict Subgroup of G st P = H99 & P
      is_Sylow_p-subgroup_of_prime p;
      consider Q9 be strict Subgroup of G such that
A27:  Q9 = H99 and
A28:  Q9 is_Sylow_p-subgroup_of_prime p by A26;
      set N = Normalizer Q9;
      now
        let y be object;
        assume
A29:    y in the carrier of P;
        ex h being Element of G st y = h & Q9 |^ h = Q9
        proof
          set h = y;
          the carrier of P c= the carrier of G by GROUP_2:def 5;
          then reconsider h as Element of G by A29;
          reconsider h9=h as Element of P by A29;
          take h;
          thus y = h;
          dom(T^h9) = E by FUNCT_2:def 1;
          then Q9 in dom(T^h9) by A27,A28;
          then reconsider Q1=Q9 as Element of E;
A30:      ex Q2 be Element of E, H1,H2 be strict Subgroup of G, g be
Element of G st Q2=the_left_translation_of(h9,p).Q1 & Q1=H1 & Q2=H2 & h9"= g &
          H2 = H1 |^ g by Def20;
          Q9 = (T^h9).Q9 & T.h9 = the_left_translation_of(h9,p) by A24,A25,A27
,Def21;
          then Q9 |^ h" = Q9 by A30,GROUP_2:48;
          then h"*h = 1_G & Q9 |^ (h"*h) = Q9 |^ h by GROUP_1:def 5,GROUP_3:60;
          hence thesis by GROUP_3:61;
        end;
        then y in N by GROUP_3:134;
        hence y in the carrier of N;
      end;
      then
A31:  the carrier of P c= the carrier of N;
A32:  now
        let y be object;
        assume
A33:    y in the carrier of Q9;
        ex h being Element of G st y = h & Q9 |^ h = Q9
        proof
          set h = y;
          the carrier of Q9 c= the carrier of G by GROUP_2:def 5;
          then reconsider h as Element of G by A33;
          take h;
          thus y = h;
          for g being Element of G holds g in Q9 iff g in Q9 |^ h
          proof
            let g be Element of G;
            hereby
              assume
A34:          g in Q9;
              ex g9 being Element of G st g = g9 |^ h & g9 in Q9
              proof
                set g9 = h * g * h";
                take g9;
                thus g9 |^ h = h" * g9 * h by GROUP_3:def 2
                  .= h" * (h * (g * h")) * h by GROUP_1:def 3
                  .= (h" * h) * (g * h") * h by GROUP_1:def 3
                  .= 1_G * (g * h") * h by GROUP_1:def 5
                  .= g * h" * h by GROUP_1:def 4
                  .= g * (h" * h) by GROUP_1:def 3
                  .= g * 1_G by GROUP_1:def 5
                  .= g by GROUP_1:def 4;
                h in Q9 by A33;
                then h" in Q9 & h * g in Q9 by A34,GROUP_2:50,51;
                hence thesis by GROUP_2:50;
              end;
              hence g in Q9 |^ h by GROUP_3:58;
            end;
            assume g in Q9 |^ h;
            then consider g9 be Element of G such that
A35:        g = g9 |^ h and
A36:        g9 in Q9 by GROUP_3:58;
A37:        h in Q9 by A33;
            then h" in Q9 by GROUP_2:51;
            then h" * g9 in Q9 by A36,GROUP_2:50;
            then h" * g9 * h in Q9 by A37,GROUP_2:50;
            hence thesis by A35,GROUP_3:def 2;
          end;
          hence thesis;
        end;
        then y in N by GROUP_3:134;
        hence y in the carrier of N;
      end;
      then
A38:  the carrier of Q9 c= the carrier of N;
      reconsider N as finite Group;
      reconsider Q99=Q9 as strict Subgroup of N by A38,GROUP_2:57;
A39:  Q99 is_Sylow_p-subgroup_of_prime p by A28,Lm9;
      reconsider P99=P9 as strict Subgroup of N by A31,GROUP_2:57;
      P99 is_Sylow_p-subgroup_of_prime p by A2,Lm9;
      then P99,Q99 are_conjugated by A39,Th12;
      then consider n be Element of N such that
A40:  P99 = Q99 |^ n by GROUP_3:def 11;
      the carrier of Q99 |^ n c= the carrier of N by GROUP_2:def 5;
      then
A41:  (the multF of G)||the carrier of Q99 |^ n = ((the multF of G)||the
      carrier of N)||the carrier of Q99 |^ n by RELAT_1:74,ZFMISC_1:96
        .= (the multF of N)||the carrier of Q99 |^ n by GROUP_2:def 5;
      n in Normalizer Q9;
      then consider n9 be Element of G such that
A42:  n = n9 and
A43:  Q9 |^ n9 = Q9 by GROUP_3:134;
A44:  now
        let y be object;
        hereby
          assume y in the carrier of Q9 |^ n9;
          then y in carr(Q9) |^ n9 by GROUP_3:def 6;
          then consider q9 be Element of G such that
A45:      y = q9 |^ n9 and
A46:      q9 in carr Q9 by GROUP_3:41;
          reconsider q99=q9 as Element of N by A32,A46;
          n9" = n" by A42,GROUP_2:48;
          then n9" * q9 = n" * q99 by GROUP_2:43;
          then
A47:      n9" * q9 * n9 = n" * q99 * n by A42,GROUP_2:43;
          q9 |^ n9 = n9" * q9 * n9 by GROUP_3:def 2
            .= q99 |^ n by A47,GROUP_3:def 2;
          then y in carr(Q99) |^ n by A45,A46,GROUP_3:41;
          hence y in the carrier of Q99 |^ n by GROUP_3:def 6;
        end;
        assume y in the carrier of Q99 |^ n;
        then y in carr(Q99) |^ n by GROUP_3:def 6;
        then consider q99 be Element of N such that
A48:    y = q99 |^ n and
A49:    q99 in carr Q99 by GROUP_3:41;
        the carrier of Q99 c= the carrier of G by GROUP_2:def 5;
        then reconsider q9=q99 as Element of G by A49;
        n9" = n" by A42,GROUP_2:48;
        then n9" * q9 = n" * q99 by GROUP_2:43;
        then
A50:    n9" * q9 * n9 = n" * q99 * n by A42,GROUP_2:43;
        q9 |^ n9 = n9" * q9 * n9 by GROUP_3:def 2
          .= q99 |^ n by A50,GROUP_3:def 2;
        then y in carr(Q9) |^ n9 by A48,A49,GROUP_3:41;
        hence y in the carrier of Q9 |^ n9 by GROUP_3:def 6;
      end;
      then the carrier of Q9 |^ n9 = the carrier of Q99 |^ n by TARSKI:2;
      then the multF of Q9 |^ n9 = (the multF of G)||the carrier of Q99 |^ n
      by GROUP_2:def 5
        .= the multF of Q99 |^ n by A41,GROUP_2:def 5;
      hence x=P9 by A23,A25,A27,A40,A43,A44,TARSKI:2;
    end;
    assume x=P9;
    then x in {x9 where x9 is Element of E: x9 is_fixed_under T} by A22;
    hence thesis by Def13;
  end;
  then
A51: the_fixed_points_of T = {P9} by TARSKI:def 1;
  card E mod p = card the_fixed_points_of T mod p by A3,Th9
    .= 1 mod p by A51,CARD_1:30;
  hence card the_sylow_p-subgroups_of_prime(p,G) mod p = 1 by A1,NAT_D:14;
A52: card the_orbit_of(P9,T9) = Index K by Th8;
  reconsider K as Subgroup of G9;
  card G = card G9 * 1;
  then
A53: card G = card K * index K by GROUP_2:147;
  ex B being finite set st B = Left_Cosets K & index K = card B by GROUP_2:146;
  then card E = index K by A52,A4,TARSKI:2;
  hence thesis by A53,NAT_D:def 3;
end;
