theorem
  for H being strict Subgroup of G holds (1).G "\/" H = H & H "\/" (1).G = H
proof
  let H be strict Subgroup of G;
  1_G in H by GROUP_2:46;
  then 1_G in carr H by STRUCT_0:def 5;
  then {1_G} c= carr H by ZFMISC_1:31;
  then
A1: carr(1).G = {1_G} & {1_G} \/ carr H = carr H by GROUP_2:def 7,XBOOLE_1:12;
  hence (1).G "\/" H = H by Th31;
  thus thesis by A1,Th31;
end;
