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 G being strict Group, a being Element of G holds a in Phi(G) iff a
  is non generating
proof
  let G be strict Group, a be Element of G;
  thus a in Phi(G) implies a is non generating
  proof
    assume a in Phi(G);
    then a in the carrier of Phi(G) by STRUCT_0:def 5;
    then a in {b where b is Element of G: b is non generating} by Th41;
    then ex b being Element of G st a = b & b is non generating;
    hence thesis;
  end;
  assume a is non generating;
  then a in {b where b is Element of G : b is non generating};
  then a in the carrier of Phi(G) by Th41;
  hence thesis by STRUCT_0:def 5;
end;
