reserve x,y,X,Y for set,
  k,l,n for Nat,
  i,i1,i2,i3,j for Integer,
  G for Group,
  a,b,c,d for Element of G,
  A,B,C for Subset of G,
  H,H1,H2, H3 for Subgroup of G,
  h for Element of H,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

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;
