reserve i,j,n,n1,n2,m,k,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat;

theorem Th15:
  Py(a,k) = Py(a,n) implies k = n
proof
  assume Py(a,k) = Py(a,n);
  then k >= n >= k by Th14;
  hence thesis by XXREAL_0:1;
end;
