theorem
  for G being strict Group, H being strict Subgroup of G holds
  Right_Cosets H = {the carrier of G} implies H = G
proof
  let G be strict Group, H be strict Subgroup of G;
  assume Right_Cosets H = {the carrier of G};
  then
A1: the carrier of G in Right_Cosets H by TARSKI:def 1;
  then reconsider T = the carrier of G as Subset of G;
  consider a being Element of G such that
A2: T = H * a by A1,Def16;
  now
    let g be Element of G;
    ex b being Element of G st g * a = b * a & b in H by A2,Th104;
    hence g in H by GROUP_1:6;
  end;
  hence thesis by Th62;
end;
