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 Th2:
  (2*a1+1)|^2 + (2*a1|^2 + 2*a1)|^2 = (2*a1|^2 + 2*a1 + 1)|^2
  proof
    ((2*a1|^2) + (2*a1) +1)|^2 =
    ((2*a1|^2) + (2*a1) +1)*((2*a1|^2) + (2*a1) +1) by NEWTON:81
    .= (2*a1|^2)*(2*a1|^2)+ (2*a1)*(2*a1) +1
    + 2*(2*a1)*(2*a1|^2) + 2*(2*a1|^2) + 2*(2*a1)
    .= (2*a1|^2)|^2 + (2*a1)*(2*a1) + 1 +
    2*(2*a1)*(2*a1|^2) + 2*(2*a1|^2) + 2*(2*a1) by NEWTON:81
    .= (2*a1|^2)|^2 + (2*a1)|^2 + 1 + 2*(2*a1|^2)*(2*a1)
    + 2*(2*a1|^2)+ 2*(2*a1)*1 by NEWTON:81
    .= ((2*a1|^2)|^2 + 2*(2*a1|^2)*(2*a1) + (2*a1)|^2)
    + ((2*2*a1|^2)+ 2*(2*a1)*1 + 1|^2)
    .= ((2*a1|^2) + (2*a1))|^2 + (2*2*a1|^2+ 2*(2*a1)*1 + 1|^2) by Lm10
    .= ((2*a1|^2) + (2*a1))|^2 + (2|^2*a1|^2+ 2*(2*a1)*1 + 1|^2) by NEWTON:81
    .= ((2*a1|^2) + (2*a1))|^2 + ((2*a1)|^2+ 2*(2*a1)*1 + 1|^2) by NEWTON:7;
    hence thesis by Lm10;
  end;
