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 Th135:
  for H being strict Subgroup of G holds card con_class H = Index
  Normalizer H
proof
  let H be strict Subgroup of G;
  defpred P[object,object] means
ex H1 being strict Subgroup of G st $1 = H1 & $2 = carr H1;
A1: for x being object st x in con_class H ex y being object st P[x,y]
  proof
    let x be object;
    assume x in con_class H;
    then reconsider H = x as strict Subgroup of G by Def1;
    reconsider y = carr H as set;
    take y;
    take H;
    thus thesis;
  end;
  consider f being Function such that
A2: dom f = con_class H and
A3: for x being object st x in con_class H holds P[x,f.x]
from CLASSES1:sch 1(A1);
A4: rng f = con_class carr H
  proof
    thus rng f c= con_class carr H
    proof
      let x be object;
      assume x in rng f;
      then consider y being object such that
A5:   y in dom f and
A6:   f.y = x by FUNCT_1:def 3;
      consider H1 being strict Subgroup of G such that
A7:   y = H1 and
A8:   x = carr H1 by A2,A3,A5,A6;
      H1,H are_conjugated by A2,A5,A7,Th107;
      then carr H1,carr H are_conjugated by Th113;
      hence thesis by A8;
    end;
    let x be object;
    assume x in con_class carr H;
    then consider B such that
A9: B = x and
A10: carr H,B are_conjugated;
    consider H1 being strict Subgroup of G such that
A11: the carrier of H1 = B by A10,Th93;
    B = carr H1 by A11;
    then H1,H are_conjugated by A10,Th113;
    then
A12: H1 in con_class H by Th107;
    then ex H2 being strict Subgroup of G st H1 = H2 & f.H1 = carr H2 by A3;
    hence thesis by A2,A9,A11,A12,FUNCT_1:def 3;
  end;
  f is one-to-one
  proof
    let x,y be object;
    assume that
A13: x in dom f & y in dom f and
A14: f.x = f.y;
    (ex H1 being strict Subgroup of G st x = H1 & f.x = carr H1 )& ex H2
    being strict Subgroup of G st y = H2 & f.y = carr H2 by A2,A3,A13;
    hence thesis by A14,GROUP_2:59;
  end;
  then con_class H,con_class carr H are_equipotent by A2,A4,WELLORD2:def 4;
  hence card con_class H = card con_class carr H by CARD_1:5
    .= Index Normalizer H by Th130;
end;
