reserve n for Nat,
  a,b,c,d for Real,
  s for Real_Sequence;

theorem Th2:
  (a+b)|^3 = a|^3+3*a|^2*b+3*b|^2*a+b|^3
proof
  (a+b)|^(2+1)=((a+b)|^2)*(a+b) by NEWTON:6
    .=(a|^2+2*a*b+b|^2)*(a+b) by Lm1
    .=a|^2*a+a|^2*b+(2*a*b*a+2*a*b*b)+(b|^2*a+b|^2*b)
    .=a|^2*a+a|^2*b+(2*a*b*a+2*a*b*b)+(b|^2*a+b|^(2+1)) by NEWTON:6
    .=a|^3+a|^2*b+(2*(a*a)*b+2*a*b*b)+(b|^2*a+b|^3) by NEWTON:6
    .=a|^3+a|^2*b+(2*(a|^1*a)*b+2*a*b*b)+(b|^2*a+b|^3)
    .=a|^3+a|^2*b+(2*(a|^(1+1))*b+2*a*b*b)+(b|^2*a+b|^3) by NEWTON:6
    .=a|^3+3*a|^2*b+2*(b*b)*a+(b|^2*a+b|^3)
    .=a|^3+3*a|^2*b+2*(b|^1*b)*a+(b|^2*a+b|^3)
    .=a|^3+3*a|^2*b+2*(b|^(1+1))*a+(b|^2*a+b|^3) by NEWTON:6
    .=a|^3+3*a|^2*b+3*b|^2*a+b|^3;
  hence thesis;
end;
