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 Gc being strict finite cyclic Group st ex k st card Gc = 2*k holds
  ex H being Subgroup of Gc st card H = 2 & H is cyclic Group
proof
  let Gc be strict finite cyclic Group;
  set n = card Gc;
  assume ex k st n=2*k;
  then consider g1 being Element of Gc such that
A1: ord g1 = 2 and
  for g2 being Element of Gc st ord g2=2 holds g1=g2 by Th24;
  take gr {g1};
  thus thesis by A1,Th4,GR_CY_1:7;
end;
