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
  b|^2 = (2*a+t)*t implies ex c st a|^2 + b|^2 = c|^2
  proof
    assume b|^2 = (2*a+t)*t; then
    A1: b|^2 = ((a+t)+ a)*((a+t)-a)
    .= (a+t)|^2 - a|^2 by NEWTON01:1;
    |.a+t.| in NAT by INT_1:3; then
    consider c such that
    A2: c =|.a+t.|;
    c|^2 = (a+t)|^2 or c|^2 = (-(a+t))|^2 by A2,ABSVALUE:1; then
    c|^2 = (a+t)|^2 by WSIERP_1:1;
    hence thesis by A1;
  end;
