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;
reserve S1,S2 for Element of Subgroups G;

theorem
  Bottom lattice G = (1).G
proof
  set L = lattice G;
  reconsider E = (1).G as Element of L by GROUP_3:def 1;
  now
    let A be Element of L;
    reconsider H = A as strict Subgroup of G by GROUP_3:def 1;
    thus A "/\" E = SubMeet(G).(A,E) by LATTICES:def 2
      .= H /\ (1).G by Def11
      .= E by GROUP_2:85;
  end;
  hence thesis by RLSUB_2:64;
end;
