reserve n,n1,n2,k,D for Nat,
        r,r1,r2 for Real,
        x,y for Integer;

theorem
  for D be square Nat,p be positive Element of [:INT,INT:] st D > 0 holds
     not p is Pell's_solution of D
  proof
    let n be square Nat;
    consider m be Nat such that
    A1:n=m^2 by PYTHTRIP:def 3;
    let p be positive Element of [:INT,INT:];set p1=p`1,p2=p`2;
    assume A2:n >0 & p is Pell's_solution of n;
    then p1^2 - n * p2^2 = 1 by Def1;
    then A3:(p1 -m*p2)*(p1+m*p2)=1 by A1;
    per cases by A3,INT_1:9;
    suppose A4:  p1 -m*p2=1 & p1+m*p2=1;
      m*p2 >= 1*m by NAT_1:14, XREAL_1:64;
      hence contradiction by A4, A1, A2;
    end;
    suppose p1 -m*p2=-1 & p1+m*p2=-1;
      hence contradiction;
    end;
  end;
