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 Th132:
  card con_class a = Index Normalizer{a}
proof
  deffunc F(object) = {$1};
  consider f being Function such that
A1: dom f = con_class a and
A2: for x being object st x in con_class a holds f.x = F(x)
from FUNCT_1:sch 3;
A3: rng f = con_class{a}
  proof
    thus rng f c= con_class{a}
    proof
      let x be object;
      assume x in rng f;
      then consider y being object such that
A4:   y in dom f and
A5:   f.y = x by FUNCT_1:def 3;
      reconsider y as Element of G by A1,A4;
      f.y = {y} by A1,A2,A4;
      then x in {{d} : d in con_class a} by A1,A4,A5;
      hence thesis by Th100;
    end;
    let x be object;
    assume x in con_class{a};
    then x in {{b} : b in con_class a} by Th100;
    then consider b such that
A6: x = {b} and
A7: b in con_class a;
    f.b = {b} by A2,A7;
    hence thesis by A1,A6,A7,FUNCT_1:def 3;
  end;
  f is one-to-one
  proof
    let x,y be object;
    assume that
A8: x in dom f & y in dom f and
A9: f.x = f.y;
    f.x = {x} & f.y = {y} by A1,A2,A8;
    hence thesis by A9,ZFMISC_1:3;
  end;
  then con_class a,con_class{a} are_equipotent by A1,A3,WELLORD2:def 4;
  hence card con_class a = card con_class{a} by CARD_1:5
    .= Index Normalizer{a} by Th130;
end;
