theorem Th154:
  for G being finite Group, H being strict Subgroup of G holds
  index H = card G implies H = (1).G
proof
  let G be finite Group, H be strict Subgroup of G;
  assume index H = card G;
  then 1 * card G = card H * card G by Th147;
  hence thesis by Th70,XCMPLX_1:5;
end;
