reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;
reserve L for Subset of Subgroups G;
reserve N2 for normal Subgroup of G;

theorem Th130:
  card con_class A = Index Normalizer A
proof
  defpred P[object,object] means
ex a st $1 = A |^ a & $2 = Normalizer A * a;
A1: for x being object st x in con_class A ex y being object st P[x,y]
  proof
    let x be object;
    assume x in con_class A;
    then consider B such that
A2: x = B and
A3: A,B are_conjugated;
    consider g such that
A4: B = A |^ g by A3,Th88;
    reconsider y = Normalizer A * g as set;
    take y;
    take g;
    thus thesis by A2,A4;
  end;
  consider f being Function such that
A5: dom f = con_class A and
A6: for x being object st x in con_class A holds P[x,f.x]
from CLASSES1:sch 1(A1);
A7: for x,y1,y2 st x in con_class A & P[x,y1] & P[x,y2] holds y1 = y2
  proof
    let x,y1,y2;
    assume x in con_class A;
    given a such that
A8: x = A |^ a and
A9: y1 = Normalizer A * a;
    given b such that
A10: x = A |^ b and
A11: y2 = Normalizer A * b;
    A = A |^ b |^ a" by A8,A10,Th54
      .= A |^ (b * a") by Th47;
    then b * a" in Normalizer A by Th129;
    hence thesis by A9,A11,GROUP_2:120;
  end;
A12: rng f = Right_Cosets Normalizer A
  proof
    thus rng f c= Right_Cosets Normalizer A
    proof
      let x be object;
      assume x in rng f;
      then consider y being object such that
A13:  y in dom f & f.y = x by FUNCT_1:def 3;
      ex a st y = A |^ a & x = Normalizer A * a by A5,A6,A13;
      hence thesis by GROUP_2:def 16;
    end;
    let x be object;
    assume x in Right_Cosets Normalizer A;
    then consider a such that
A14: x = Normalizer A * a by GROUP_2:def 16;
    set y = A |^ a;
    A,A |^ a are_conjugated by Th88;
    then
A15: y in con_class A;
    then ex b st y = A |^ b & f.y = Normalizer A * b by A6;
    then x = f.y by A7,A14,A15;
    hence thesis by A5,A15,FUNCT_1:def 3;
  end;
  f is one-to-one
  proof
    let x,y be object;
    assume that
A16: x in dom f and
A17: y in dom f and
A18: f.x = f.y;
    consider b such that
A19: y = A |^ b and
A20: f.y = Normalizer A * b by A5,A6,A17;
    consider a such that
A21: x = A |^ a and
A22: f.x = Normalizer A * a by A5,A6,A16;
    b * a" in Normalizer A by A18,A22,A20,GROUP_2:120;
    then ex h st b * a" = h & A |^ h = A by Th129;
    then A = A |^ b |^ a" by Th47;
    hence thesis by A21,A19,Th54;
  end;
  then con_class A,Right_Cosets Normalizer A are_equipotent by A5,A12,
WELLORD2:def 4;
  hence card con_class A = card Right_Cosets Normalizer A by CARD_1:5
    .= Index Normalizer A by GROUP_2:145;
end;
