reserve a,b,m,x,y,i1,i2,i3,i for Integer,
  k,p,q,n for Nat,
  c,c1,c2 for Element of NAT,
  z for set;

theorem
  n>0 & p is prime & p,q are_coprime implies not ex x being
  Integer st (p*x^2 - q) mod p|^n = 0
proof
  assume that
A1: n>0 and
A2: p is prime and
A3: p,q are_coprime;
  given x such that
A4: (p*x^2 - q) mod p|^n = 0;
  (p*x^2) mod p|^n = q mod p|^n by A2,A4,Th22;
  then p*(x^2 mod p|^(n-'1)) = q mod p|^n by A1,A2,Th19;
  then p divides q mod p|^n;
  hence contradiction by A1,A2,A3,Th21;
end;
