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 Th10:
  for G being finite Group, a being Element of G holds card gr {a
  |^s} = card gr {a|^k} & a|^k in gr {a|^s} implies gr {a|^s} = gr {a|^k}
proof
  let G be finite Group, a be Element of G;
  assume
A1: card gr {a|^s} = card gr {a|^k};
  assume a|^k in gr {a|^s};
  then reconsider h=a|^k as Element of gr {a|^s} by STRUCT_0:def 5;
  card gr {h} = card gr {a|^s} by A1,Th3;
  hence gr {a|^s} = gr {h} by GROUP_2:73
    .= gr {a|^k} by Th3;
end;
