reserve X for set;

theorem Th9:
  for G being Group
  for K being trivial Subgroup of G
  for H being Subgroup of G
  st H is Subgroup of K
  holds H is trivial Subgroup of G
proof
  let G be Group;
  let K be trivial Subgroup of G;
  let H be Subgroup of G;
  assume A1: H is Subgroup of K;
  the carrier of H = {1_G}
  proof
    the multMagma of K = (1).G by Th7;
    then the carrier of K = {1_G} by GROUP_2:def 7;
    then B1: the carrier of H c= {1_G} by A1,GROUP_2:def 5;
    (1).G is Subgroup of H by GROUP_2:65;
    then the carrier of (1).G c= the carrier of H by GROUP_2:def 5;
    then {1_G} c= the carrier of H by GROUP_2:def 7;
    hence the carrier of H = {1_G} by B1,XBOOLE_0:def 10;
  end;
  hence H is trivial Subgroup of G;
end;
