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 Th8:
  for G being finite Group, a being Element of G holds G=gr {a} &
  card G = n & n = p * s implies ord (a|^p) = s
proof
  let G be finite Group, a be Element of G;
  assume that
A1: G=gr {a} and
A2: card G = n and
A3: n = p * s;
A4: a|^p is not being_of_order_0 & s <> 0 by A2,A3,GR_CY_1:6;
A5: p <> 0 by A2,A3;
A6: for k being Nat st a|^p|^ k = 1_G & k <> 0 holds s <= k
  proof
    let k be Nat;
    assume that
A7: a|^p|^k=1_G and
A8: k <> 0 & s > k;
A9: a|^(p*k) = 1_G by A7,GROUP_1:35;
A10: ord a = n & a is not being_of_order_0 by A1,A2,GR_CY_1:6,7;
    p*s > p*k & p*k <> 0 by A5,A8,XCMPLX_1:6,XREAL_1:68;
    hence contradiction by A3,A10,A9,GROUP_1:def 11;
  end;
  a|^p|^s = a|^n by A3,GROUP_1:35
    .=1_G by A2,GR_CY_1:9;
  hence thesis by A4,A6,GROUP_1:def 11;
end;
