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 Th38:
  for G being Group holds (ex H being strict Subgroup of G st H is
  maximal) implies (a in Phi(G) iff for H being strict Subgroup of G st H is
  maximal holds a in H)
proof
  let G be Group;
  set X = {A where A is Subset of G: ex K being strict Subgroup of G st A =
  the carrier of K & K is maximal};
  assume
A1: ex H being strict Subgroup of G st H is maximal;
  then consider H being strict Subgroup of G such that
A2: H is maximal;
  thus a in Phi(G) implies for H being strict Subgroup of G st H is maximal
  holds a in H
  proof
    assume a in Phi(G);
    then a in the carrier of Phi(G) by STRUCT_0:def 5;
    then
A3: a in meet X by A1,Def7;
    let H be strict Subgroup of G;
    assume H is maximal;
    then carr H in X;
    then a in carr H by A3,SETFAM_1:def 1;
    hence thesis by STRUCT_0:def 5;
  end;
  assume
A4: for H being strict Subgroup of G st H is maximal holds a in H;
A5: now
    let Y;
    assume Y in X;
    then consider A being Subset of G such that
A6: Y = A and
A7: ex H being strict Subgroup of G st A = the carrier of H & H is maximal;
    consider H being strict Subgroup of G such that
A8: A = the carrier of H and
A9: H is maximal by A7;
    a in H by A4,A9;
    hence a in Y by A6,A8,STRUCT_0:def 5;
  end;
  carr H in X by A2;
  then a in meet X by A5,SETFAM_1:def 1;
  then a in the carrier of Phi(G) by A1,Def7;
  hence thesis by STRUCT_0:def 5;
end;
