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|^2 + b|^2 = c|^2 implies ex t st b|^2 = (2*a+t)*t
  proof
    assume a|^2+b|^2 = c|^2; then
    b|^2 = c|^2 - a|^2
    .= (c-a)*(c+a) by NEWTON01:1
    .= (c-a)*((c-a)+2*a);
    hence thesis;
  end;
