
theorem Th8:
  for G being Group, E being non empty set, a being Element of E,
  T being LeftOperation of G, E holds card the_orbit_of(a,T) = Index
  the_strict_stabilizer_of(a,T)
proof
  let G be Group;
  let E be non empty set;
  let a be Element of E;
  let T be LeftOperation of G, E;
  set H = the_strict_stabilizer_of(a,T);
  set f = {[b,A] where b is Element of E, A is Subset of G: ex g being Element
  of G st b = (T^g).a & A = g * H};
  reconsider A9 = {a} as Subset of E;
A1: now
    let x be object;
    assume x in f;
    then consider b be Element of E, A be Subset of G such that
A2: [b,A]=x and
    ex g being Element of G st b = (T^g).a & A = g * H;
    reconsider b,A as object;
    take b,A;
    thus x=[b,A] by A2;
  end;
  now
    let x,y1,y2 be object;
    assume [x,y1] in f;
    then consider b1 be Element of E, A1 be Subset of G such that
A3: [b1,A1]=[x,y1] and
A4: ex g being Element of G st b1 = (T^g).a & A1 = g * H;
    assume [x,y2] in f;
    then consider b2 be Element of E, A2 be Subset of G such that
A5: [b2,A2]=[x,y2] and
A6: ex g being Element of G st b2 = (T^g).a & A2 = g * H;
A7: y2=A2 by A5,XTUPLE_0:1;
    consider g1 be Element of G such that
A8: b1 = (T^g1).a and
A9: A1 = g1 * H by A4;
    consider g2 be Element of G such that
A10: b2 = (T^g2).a and
A11: A2 = g2 * H by A6;
    x=b2 by A5,XTUPLE_0:1;
    then dom(T^g1)=E & (T^g1).a = (T^g2).a by A3,A8,A10,FUNCT_2:def 1
,XTUPLE_0:1;
    then dom(T^g2)=E & Im(T^g1,a) = {(T^g2).a} by FUNCT_1:59,FUNCT_2:def 1;
    then
A12: Im(T^g1,a) = Im(T^g2,a) by FUNCT_1:59;
    set g = g2"*g1;
    reconsider g as Element of G;
A13: the carrier of the_strict_stabilizer_of(A9,T) = {g9 where g9 is
    Element of G: (T^g9) .: A9 = A9} by Def10;
    (T^g) .: A9 = (T^g2" * (T^g1)) .: A9 by Def1
      .= (T^g2").:((T^g1) .: A9) by RELAT_1:126
      .= (T^g2" *(T^g2)) .: A9 by A12,RELAT_1:126
      .= (T^(g2"*g2)) .: A9 by Def1
      .= (T^1_G) .: A9 by GROUP_1:def 5
      .= (id E) .: A9 by Def1
      .= A9 by FUNCT_1:92;
    then g in {g9 where g9 is Element of G: (T^g9) .: A9 = A9};
    then
A14: g in the_strict_stabilizer_of(A9,T) by A13;
    y1=A1 by A3,XTUPLE_0:1;
    hence y1=y2 by A7,A9,A11,A14,GROUP_2:114;
  end;
  then reconsider f as Function by A1,FUNCT_1:def 1,RELAT_1:def 1;
  for y being object holds y in Left_Cosets H iff
    ex x being object st [x,y] in f
  proof
    let y be object;
    hereby
      assume
A15:  y in Left_Cosets H;
      then reconsider A=y as Subset of G;
      consider g be Element of G such that
A16:  A = g * H by A15,GROUP_2:def 15;
      reconsider x=(T^g).a as object;
      take x;
      thus [x,y] in f by A16;
    end;
    given x be object such that
A17: [x,y] in f;
    consider b be Element of E, A be Subset of G such that
A18: [b,A]=[x,y] and
A19: ex g being Element of G st b = (T^g).a & A = g * H by A17;
    A=y by A18,XTUPLE_0:1;
    hence thesis by A19,GROUP_2:def 15;
  end;
  then
A20: rng f = Left_Cosets H by XTUPLE_0:def 13;
  now
    let x1,x2 be object;
A21: the carrier of the_strict_stabilizer_of(A9,T) = {g9 where g9 is
    Element of G: (T^g9) .: A9 = A9} by Def10;
    assume x1 in dom f;
    then [x1,f.x1] in f by FUNCT_1:1;
    then consider b1 be Element of E, A1 be Subset of G such that
A22: [b1,A1]=[x1,f.x1] and
A23: ex g being Element of G st b1 = (T^g).a & A1=g * H;
    assume x2 in dom f;
    then [x2,f.x2] in f by FUNCT_1:1;
    then consider b2 be Element of E, A2 be Subset of G such that
A24: [b2,A2]=[x2,f.x2] and
A25: ex g being Element of G st b2 = (T^g).a & A2=g * H;
    consider g2 be Element of G such that
A26: b2 = (T^g2).a and
A27: A2 = g2 * H by A25;
    assume
A28: f.x1 = f.x2;
    consider g1 be Element of G such that
A29: b1 = (T^g1).a and
A30: A1 = g1 * H by A23;
    set g = g2"*g1;
    f.x2=A2 by A24,XTUPLE_0:1;
    then g1 * H = g2 * H by A22,A30,A27,A28,XTUPLE_0:1;
    then g in H by GROUP_2:114;
    then g in the carrier of the_strict_stabilizer_of(A9,T);
    then
A31: ex g9 being Element of G st g=g9 & (T^g9) .: A9 = A9 by A21;
    (T^g1).:A9 = (id E).:((T^g1).:A9) by FUNCT_1:92
      .= (T^1_G).:((T^g1).:A9) by Def1
      .= (T^(g2*g2")).:((T^g1).:A9) by GROUP_1:def 5
      .= (T^(g2*g2")*(T^g1)).:A9 by RELAT_1:126
      .= (T^((g2*g2")*g1)).:A9 by Def1
      .= (T^(g2*g)).:A9 by GROUP_1:def 3
      .= ((T^g2)*(T^g)).:A9 by Def1
      .= (T^g2).:A9 by A31,RELAT_1:126;
    then dom(T^g2)=E & Im(T^g1,a) = Im(T^g2,a) by FUNCT_2:def 1;
    then dom(T^g1)=E & Im(T^g1,a) = {(T^g2).a} by FUNCT_1:59,FUNCT_2:def 1;
    then
A32: {(T^g1).a} = {(T^g2).a} by FUNCT_1:59;
A33: x2=b2 by A24,XTUPLE_0:1;
    x1=b1 by A22,XTUPLE_0:1;
    hence x1 = x2 by A33,A29,A26,A32,ZFMISC_1:18;
  end;
  then
A34: f is one-to-one by FUNCT_1:def 4;
  for x being object holds x in the_orbit_of(a,T) iff
    ex y being object st [x,y] in f
  proof
    let x be object;
    hereby
      assume x in the_orbit_of(a,T);
      then consider b be Element of E such that
A35:  b=x and
A36:  a,b are_conjugated_under T;
      consider g be Element of G such that
A37:  b = (T^g).a by A36;
      reconsider y=g * H as object;
      take y;
      thus [x,y] in f by A35,A37;
    end;
    given y be object such that
A38: [x,y] in f;
    consider b be Element of E, A be Subset of G such that
A39: [b,A]=[x,y] & ex g being Element of G st b = (T^g).a & A = g * H by A38;
    b=x & a,b are_conjugated_under T by A39,XTUPLE_0:1;
    hence thesis;
  end;
  then dom f = the_orbit_of(a,T) by XTUPLE_0:def 12;
  then the_orbit_of(a,T), Left_Cosets H are_equipotent by A34,A20;
  hence thesis by CARD_1:5;
end;
