reserve F,G for Group;
reserve G1 for Subgroup of G;
reserve Gc for cyclic Group;
reserve H for Subgroup of Gc;
reserve f for Homomorphism of G,Gc;
reserve a,b for Element of G;
reserve g for Element of Gc;
reserve a1 for Element of G1;
reserve k,m,n,p,s for Element of NAT;
reserve i0,i,i1,i2 for Integer;
reserve j,j1 for Element of INT.Group;
reserve x,y,t for set;

theorem Th5:
  for G being strict Group,b being Element of G holds ( for a
  being Element of G holds ex i st a=b|^i ) iff G = gr {b}
proof
  let G be strict Group;
  let b be Element of G;
  thus ( for a being Element of G holds ex i st a=b|^i ) implies G = gr {b}
  proof
    assume
A1: for a being Element of G holds ex i st a=b|^i;
    for a being Element of G holds a in gr {b}
    proof
      let a be Element of G;
      ex i st a=b|^i by A1;
      hence thesis by GR_CY_1:5;
    end;
    hence thesis by GROUP_2:62;
  end;
  assume
A2: G= gr {b};
  let a be Element of G;
  a in gr {b} by A2,STRUCT_0:def 5;
  hence thesis by GR_CY_1:5;
end;
