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 Th31:
  for H being strict Subgroup of G holds gr carr H = H
proof
  let H be strict Subgroup of G;
  for H1 be strict Subgroup of G st carr H c= the carrier of H1 holds H is
  Subgroup of H1 by GROUP_2:57;
  hence thesis by Def4;
end;
