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

theorem
  (a+b+c)|^3=a|^3+b|^3+c|^3+(3*a|^2*b+3*a|^2*c)+ (3*b|^2*a+3*b|^2*c)+(3*
  c|^2*a+3*c|^2*b)+6*a*b*c
proof
  (a+b+c)|^(2+1)=((a+b+c)|^2)*(a+b+c) by NEWTON:6
    .=(a|^2+b|^2+c|^2+2*a*b+2*a*c+2*b*c)*(a+b+c) by Th1
    .=(a|^2*a+a|^2*b+a|^2*c)+(b|^2*a+b|^2*b+b|^2*c) +(c|^2*a+c|^2*b+c|^2*c)+
(2*a*b*a+2*a*b*b+2*a*b*c) +(2*a*c*a+2*a*c*b+2*a*c*c)+(2*b*c*a+2*b*c*b+2*b*c*c)
    .=(a|^(2+1)+a|^2*b+a|^2*c)+(b|^2*a+b|^2*b+b|^2*c) +(c|^2*a+c|^2*b+c|^2*c
)+(2*a*b*a+2*a*b*b+2*a*b*c) +(2*a*c*a+2*a*c*b+2*a*c*c)+(2*b*c*a+2*b*c*b+2*b*c*c
  ) by NEWTON:6
    .=(a|^3+a|^2*b+a|^2*c)+(b|^2*a+b|^(2+1)+b|^2*c) +(c|^2*a+c|^2*b+c|^2*c)+
(2*a*b*a+2*a*b*b+2*a*b*c) +(2*a*c*a+2*a*c*b+2*a*c*c)+(2*b*c*a+2*b*c*b+2*b*c*c)
  by NEWTON:6
    .=(a|^3+a|^2*b+a|^2*c)+(b|^2*a+b|^3+b|^2*c)+(c|^2*a+c|^2*b+c|^3)+ (2*(a*
a)*b+2*a*(b*b)+2*a*b*c)+(2*(a*a)*c+2*a*c*b+2*a*(c*c)) +(2*b*c*a+2*(b*b)*c+2*b*(
  c*c)) by NEWTON:6
    .=(a|^3+a|^2*b+a|^2*c)+(b|^2*a+b|^3+b|^2*c)+(c|^2*a+c|^2*b+c|^3)+ (2*(a
|^1*a)*b+2*a*(b*b)+2*a*b*c)+(2*(a*a)*c+2*a*c*b+2*a*(c*c)) +(2*b*c*a+2*(b*b)*c+2
  *b*(c*c))
    .=(a|^3+a|^2*b+a|^2*c)+(b|^2*a+b|^3+b|^2*c)+(c|^2*a+c|^2*b+c|^3)+ (2*(a
|^(1+1))*b+2*a*(b*b)+2*a*b*c)+(2*(a*a)*c+2*a*c*b+2*a*(c*c)) +(2*b*c*a+2*(b*b)*c
  +2*b*(c*c)) by NEWTON:6
    .=(a|^3+a|^2*b+a|^2*c)+(b|^2*a+b|^3+b|^2*c)+(c|^2*a+c|^2*b+c|^3)+ (2*a|^
2*b+2*a*(b|^1*b)+2*a*b*c)+(2*(a*a)*c+2*a*c*b+2*a*(c*c)) +(2*b*c*a+2*(b*b)*c+2*b
  *(c*c))
    .=(a|^3+a|^2*b+a|^2*c)+(b|^2*a+b|^3+b|^2*c)+(c|^2*a+c|^2*b+c|^3)+ (2*a|^
2*b+2*a*b|^(1+1)+2*a*b*c)+(2*(a*a)*c+2*a*c*b+2*a*(c*c)) +(2*b*c*a+2*(b*b)*c+2*b
  *(c*c)) by NEWTON:6
    .=(a|^3+a|^2*b+a|^2*c)+(b|^2*a+b|^3+b|^2*c)+(c|^2*a+c|^2*b+c|^3)+ (2*a|^
2*b+2*a*b|^2+2*a*b*c)+(2*(a|^2)*c+2*a*c*b+2*a*(c*c)) +(2*b*c*a+2*(b*b)*c+2*b*(c
  *c)) by WSIERP_1:1
    .=(a|^3+a|^2*b+a|^2*c)+(b|^2*a+b|^3+b|^2*c)+(c|^2*a+c|^2*b+c|^3)+ (2*a|^
2*b+2*a*b|^2+2*a*b*c)+(2*a|^2*c+2*a*c*b+2*a*(c|^2)) +(2*b*c*a+2*(b*b)*c+2*b*(c*
  c)) by WSIERP_1:1
    .=(a|^3+a|^2*b+a|^2*c)+(b|^2*a+b|^3+b|^2*c)+(c|^2*a+c|^2*b+c|^3)+ (2*a|^
2*b+2*a*b|^2+2*a*b*c)+(2*a|^2*c+2*a*c*b+2*a*(c|^2)) +(2*b*c*a+2*(b*b)*c+2*b*(c
  |^2)) by WSIERP_1:1
    .=(a|^3+b|^3+c|^3+a|^2*b+a|^2*c)+(b|^2*a+b|^2*c)+(c|^2*a+c|^2*b)+ (2*a|^
2*b+2*a*b|^2+2*a*b*c)+(2*a|^2*c+2*a*c*b+2*a*c|^2) +(2*b*c*a+2*b|^2*c+2*(c|^2)*b
  ) by WSIERP_1:1
    .=(a|^3+b|^3+c|^3)+(3*a|^2*b+3*a|^2*c)+ (3*b|^2*a+3*b|^2*c)+(3*c|^2*a+3*
  c|^2*b)+6*a*b*c;
  hence thesis;
end;
