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 Th9:
  s divides k implies a|^k in gr {a|^s}
proof
  assume s divides k;
  then consider p be Nat such that
A1: k=s*p by NAT_D:def 3;
  ex i st a|^k=a|^s|^i
  proof
    reconsider p9=p as Integer;
    take p9;
    thus thesis by A1,GROUP_1:35;
  end;
  hence thesis by GR_CY_1:5;
end;
