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 Th40:
  for G being Group, H being strict Subgroup of G holds H is
  maximal implies Phi(G) is Subgroup of H
proof
  let G be Group, H be strict Subgroup of G;
  assume H is maximal;
  then for a be Element of G holds a in Phi(G) implies a in H by Th38;
  hence thesis by GROUP_2:58;
end;
