reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem Th84:
  for n st 2*n+1 is prime for k st 2*n > k & k > 1
  holds not 2*n+1 divides a|^n - k & not 2*n+1 divides a|^n + k
  proof
    let n such that
    A1: 2*n + 1 is prime;
    set p = 2*n+1;
    2*n <> 0 by A1; then
    A1a: n > 0;
    let k such that
    A1b: 2*n > k & k > 1;
    A1c: k-1 > 1-1 by A1b,XREAL_1:9;
    A1d: (k-1)+2 > (k-1)+1 > (k-1)+0 by XREAL_1:6;
    2*n+1 > k+1 by A1b, XREAL_1:6; then
    2*n+1 > k+1 & 2*n+1 > k by A1d,XXREAL_0:2; then
    A2: 2*n+1 > k+1 & 2*n+1 > k & 2*n+1 > k-1 by A1d,XXREAL_0:2;
    A3: p divides a or p divides a|^n - 1 or p divides a|^n + 1 by A1,Th70;
    a divides a|^n by A1a,NAT_3:3; then
    p divides a implies p divides a|^n by INT_2:9; then
    A4: p divides (-1)*(a|^n) or p divides (-1)*(a|^n -1) or
      p divides (-1)*(a|^n + 1) by A3,INT_2:2;
    assume not thesis; then
    p divides (-a|^n) + (a|^n - k) or
    p divides (-a|^n) + (a|^n + k) or
    p divides (-a|^n -1) + (a|^n - k) or
    p divides (-a|^n -1) + (a|^n + k) or
    p divides (-a|^n +1) + (a|^n - k) or
    p divides (-a|^n +1) + (a|^n + k) by A4,WSIERP_1:4; then
    p divides -k or p divides k or p divides -(1+k) or
      p divides -1+k or p divides -(k-1) or p divides 1+k;
    hence contradiction by A1b,A1c,A2,INT_2:27,INT_2:10;
  end;
