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 Th26:
  for G being strict Group holds for g being Homomorphism of G,F
  st G is cyclic holds Image g is cyclic
proof
  let G be strict Group;
  let g be Homomorphism of G,F;
  assume G is cyclic;
  then consider a being Element of G such that
A1: G=gr{a};
  ex f1 being Element of Image g st Image g=gr{f1}
  proof
    g.a in Image g by GROUP_6:45;
    then reconsider f=g.a as Element of Image g by STRUCT_0:def 5;
    take f;
    for d being Element of Image g holds ex i st d=f|^i
    proof
      let d be Element of Image g;
      d in Image g by STRUCT_0:def 5;
      then consider b being Element of G such that
A2:   d=g.(b) by GROUP_6:45;
      b in gr{a} by A1,STRUCT_0:def 5;
      then consider i such that
A3:   b=a|^i by GR_CY_1:5;
      take i;
      d=(g.a)|^i by A2,A3,GROUP_6:37
        .=f|^i by GROUP_4:2;
      hence thesis;
    end;
    hence thesis by Th5;
  end;
  hence thesis;
end;
