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

theorem
  (a+b)|^4 = a|^4+4*a|^3*b+6*a|^2*b|^2+4*b|^3*a+b|^4
proof
  (a+b)|^(3+1)=((a+b)|^3)*(a+b) by NEWTON:6
    .=(a|^3+3*a|^2*b+3*b|^2*a+b|^3)*(a+b) by Th2
    .=a|^3*a+a|^3*b+(3*a|^2*b*a+3*a|^2*b*b)+(3*b|^2*a*a+3*b|^2*a*b)+ (b|^3*a
  +b|^3*b)
    .=a|^3*a+a|^3*b+(3*a|^2*b*a+3*a|^2*b*b)+(3*b|^2*a*a+3*b|^2*a*b)+ (b|^3*a
  +b|^(3+1)) by NEWTON:6
    .=a|^4+a|^3*b+(3*(a|^2*a)*b+3*a|^2*b*b)+(3*b|^2*a*a+3*b|^2*a*b)+ (b|^3*a
  +b|^4) by NEWTON:6
    .=a|^4+a|^3*b+(3*(a|^(2+1))*b+3*a|^2*b*b)+(3*b|^2*a*a+3*b|^2*a*b)+ (b|^3
  *a+b|^4) by NEWTON:6
    .=a|^4+a|^3*b+(3*a|^3*b+3*a|^2*(b*b))+(3*b|^2*a*a+3*b|^2*a*b)+ (b|^3*a+b
  |^4)
    .=a|^4+a|^3*b+(3*a|^3*b+3*a|^2*(b|^1*b))+(3*b|^2*a*a+3*b|^2*a*b)+ (b|^3*
  a+b|^4)
    .=a|^4+a|^3*b+(3*a|^3*b+3*a|^2*(b|^(1+1)))+(3*b|^2*a*a+3*b|^2*a*b)+ (b|^
  3*a+b|^4) by NEWTON:6
    .=a|^4+a|^3*b+(3*a|^3*b+3*a|^2*b|^2)+(3*b|^2*(a*a)+3*b|^2*a*b)+ (b|^3*a+
  b|^4)
    .=a|^4+a|^3*b+(3*a|^3*b+3*a|^2*b|^2)+(3*b|^2*(a|^1*a)+3*b|^2*b*a)+ (b|^3
  *a+b|^4)
    .=a|^4+a|^3*b+(3*a|^3*b+3*a|^2*b|^2)+(3*b|^2*a|^(1+1)+3*(b|^2*b)*a)+ (b
  |^3*a+b|^4) by NEWTON:6
    .=a|^4+a|^3*b+(3*a|^3*b+3*a|^2*b|^2)+(3*b|^2*a|^2+3*(b|^(2+1))*a)+ (b|^3
  *a+b|^4) by NEWTON:6
    .=a|^4+4*a|^3*b+6*a|^2*b|^2+4*b|^3*a+b|^4;
  hence thesis;
end;
