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 Th6:
  for G being strict finite Group,b being Element of G holds (for
  a being Element of G holds ex p st a=b|^p) iff G = gr {b}
proof
  let G be strict finite Group,b be Element of G;
  reconsider n1=card G as Integer;
  thus (for a being Element of G holds ex p st a=b|^p) implies G = gr {b}
  proof
    assume
A1: for a being Element of G holds ex p st a=b|^p;
    for a being Element of G holds ex i st a=b|^i
    proof
      let a be Element of G;
      consider p such that
A2:   a=b|^p by A1;
      reconsider p1=p as Integer;
      take p1;
      thus thesis by A2;
    end;
    hence thesis by Th5;
  end;
  assume
A3: G= gr {b};
  let a be Element of G;
  consider i such that
A4: a= b|^i by A3,Th5;
  reconsider p=i mod n1 as Element of NAT by INT_1:3,NEWTON:64;
  take p;
  a = b|^((i div n1) * n1 + (i mod n1)) by A4,NEWTON:66
    .= b|^((i div n1)*n1) *(b|^(i mod n1)) by GROUP_1:33
    .= b|^(i div n1)|^card G *(b|^(i mod n1)) by GROUP_1:35
    .= (1_G) *(b|^(i mod n1)) by GR_CY_1:9
    .= b|^(i mod n1) by GROUP_1:def 4;
  hence thesis;
end;
