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;

theorem Th64:
  card H = card(H |^ a)
proof
  deffunc F(Element of G) = $1 |^ a;
  consider f being Function of the carrier of G, the carrier of G such that
A1: for g holds f.g = F(g) from FUNCT_2:sch 4;
  set g = f | (the carrier of H);
A2: dom f = the carrier of G & the carrier of H c= the carrier of G by
FUNCT_2:def 1,GROUP_2:def 5;
  then
A3: dom g = the carrier of H by RELAT_1:62;
A4: rng g = the carrier of H |^ a
  proof
    thus rng g c= the carrier of H |^ a
    proof
      let x be object;
      assume x in rng g;
      then consider y being object such that
A5:   y in dom g and
A6:   g.y = x by FUNCT_1:def 3;
      reconsider y as Element of H by A2,A5,RELAT_1:62;
      reconsider y as Element of G by GROUP_2:42;
A7:   f.y = g.y by A5,FUNCT_1:47;
      f.y = y |^ a by A1;
      then x in carr H |^ a by A6,A7,Th41;
      hence thesis by Def6;
    end;
    let x be object;
    assume x in the carrier of H |^ a;
    then x in carr H |^ a by Def6;
    then consider b such that
A8: x = b |^ a and
A9: b in carr H by Th41;
A10: f.b = b |^ a by A1;
    g.b = f.b by A3,A9,FUNCT_1:47;
    hence thesis by A3,A8,A9,A10,FUNCT_1:def 3;
  end;
  g is one-to-one
  proof
    let x,y be object;
    assume that
A11: x in dom g and
A12: y in dom g and
A13: g.x = g.y;
    reconsider b = x, c = y as Element of H by A2,A11,A12,RELAT_1:62;
    reconsider b,c as Element of G by GROUP_2:42;
A14: f.x = b |^ a & f.y = c |^ a by A1;
    f.x = g.x by A11,FUNCT_1:47;
    hence thesis by A12,A13,A14,Th16,FUNCT_1:47;
  end;
  then the carrier of H,the carrier of H |^ a are_equipotent by A3,A4,
WELLORD2:def 4;
  hence thesis by CARD_1:5;
end;
