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 Th7:
  for G being strict Group,a being Element of G holds G is finite
& G = gr {a} implies for G1 being strict Subgroup of G holds ex p st G1 = gr {a
  |^p}
proof
  let G be strict Group,a be Element of G;
  assume that
A1: G is finite and
A2: G = gr {a};
  let G1 be strict Subgroup of G;
  G is cyclic Group by A2,GR_CY_1:def 7;
  then G1 is cyclic Group by A1,GR_CY_1:20;
  then consider b being Element of G1 such that
A3: G1 = gr {b} by GR_CY_1:def 7;
  reconsider b1 = b as Element of G by GROUP_2:42;
  consider p such that
A4: b1 = a|^p by A1,A2,Th6;
  take p;
  thus thesis by A3,A4,Th3;
end;
