theorem Th147:
  for G being finite Group, H being Subgroup of G holds
    card G = card H * index H
proof
  let G be finite Group, H be Subgroup of G;
  reconsider C = Left_Cosets H as finite set;
  now
    let X be set;
    assume
A1: X in C;
    then reconsider x = X as Subset of G;
    x is finite;
    then reconsider B = X as finite set;
    take B;
    thus B = X;
    consider a being Element of G such that
A2: x = a * H by A1,Def15;
    ex B,C being finite set st B = a * H & C = H * a & card H = card B &
    card H = card C by Th133;
    hence card B = card H by A2;
    let Y;
    assume that
A3: Y in C and
A4: X <> Y;
    reconsider y = Y as Subset of G by A3;
A5: ex b being Element of G st y = b * H by A3,Def15;
    hence X,Y are_equipotent by A2,Th128;
    thus X misses Y by A2,A4,A5,Th115;
  end;
  then
A6: ex D being finite set st D = union C & card D = card H * card C by Lm5;
  union Left_Cosets H = the carrier of G by Th137;
  hence thesis by A6,Def18;
end;
