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;

theorem
  a,b are_coprime & c is even & a|^n + b|^n = c|^n implies
    a+b is even & a-b is even
  proof
    assume
    A1: a,b are_coprime & c is even & a|^n + b|^n = c|^n;
    n = 0 implies 1*a|^n +1*b|^n > 1*c|^n; then
    (a is even & b is even) or (a is odd & b is odd) by A1;
    hence thesis;
  end;
