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
  for G,F being strict Group st G,F are_isomorphic & G is cyclic holds F
  is cyclic
proof
  let G,F be strict Group;
  assume that
A1: G,F are_isomorphic and
A2: G is cyclic;
  consider h being Homomorphism of G,F such that
A3: h is bijective by A1,GROUP_6:def 11;
  h is onto by A3,FUNCT_2:def 4;
  then Image h = F by GROUP_6:57;
  hence thesis by A2,Th26;
end;
