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 Th4:
  gr {a} is cyclic Group
proof
  ex a1 being Element of gr {a} st gr {a}=gr {a1}
  proof
    a in gr {a} by Th2;
    then reconsider a1=a as Element of gr {a} by STRUCT_0:def 5;
    take a1;
    thus thesis by Th3;
  end;
  hence thesis by GR_CY_1:def 7;
end;
