reserve a,b,c,d,x,j,k,l,m,n for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem
  a >= 3 iff a|^2 + a|^2 > (a+1)|^2
  proof
    a >= 3 implies a|^2 + a|^2 > (a+1)|^2
    proof
      assume a >= 3; then
      consider b such that
      A2: a = 3 + b by NAT_1:10;
      defpred P[Nat] means (3+$1)|^2 + (3+$1)|^2 > (3+$1+1)|^2;
      A3: P[0]
      proof
        3|^2 = 3*3 & 4|^2 = 4*4 by NEWTON:81;
        hence thesis;
      end;
      A4: P[k] implies P[k+1]
      proof
        assume P[k]; then
        1+(2+k) >= 1+0 & (3+k)|^2 + ((3+k)+0)|^2 > ((3+k)+0+1)|^2
        by XREAL_1:6; then
        ((3+k)+1)|^2 + ((3+k)+0+1)|^2 > ((3+k)+1+0+1)|^2 by Th47;
        hence thesis;
      end;
      for m holds P[m] from NAT_1:sch 2(A3,A4);
      hence thesis by A2;
    end;
    hence thesis by Lm120;
  end;
