reserve d,i,j,k,m,n,p,q,x,k1,k2 for Nat,
  a,c,i1,i2,i3,i5 for Integer;

theorem
  p is prime & i mod p = -1 implies i^2 mod p = 1
proof
  assume that
A1: p is prime and
A2: i mod p = -1;
A3: p > 1 by A1,INT_2:def 4;
  p <> 0 by A1,INT_2:def 4;
  then i mod p = i - (i div p)*p by INT_1:def 10;
  then i = (i div p)*p - 1 by A2;
  then i^2 = ((((i div p)*p) - 2)*(i div p))*p + 1;
  then i^2 mod p = 1 mod p by Th10
    .= 1 by A3,NAT_D:24;
  hence thesis;
end;
