reserve x,y,a,b,c,p,q for Real;
reserve m,n for Element of NAT;

theorem Th11:
  a|^3+b|^3 = (a+b)*(a^2-a*b+b^2) & a|^5+b|^5 = (a+b)*(a|^4-a|^3*b
  +a|^2*b|^2-a*b|^3+b|^4)
proof
A1: (a+b)*(a|^4-a|^3*b+a|^2*b|^2-a*b|^3+b|^4) =a|^4*a+b*a|^4+0*a|^4-(a|^3*b)
  *(a+b)+(a|^2*b|^2)*(a+b) -(a*b|^3)*(a+b)+b|^4*(a+b)
    .=a|^4*a|^1+b*a|^4+0*a|^4-(a|^3*b)*(a+b)+(a|^2*b|^2)*(a+b) -(a*b|^3)*(a+
  b)+b|^4*(a+b)
    .=a|^(4+1)+b*a|^4-(a|^3*b)*(a+b+0)+(a|^2*b|^2)*(a+b) -(a*b|^3)*(a+b)+b|^
  4*(a+b) by NEWTON:8
    .=(a|^5+b*a|^4)-(a*a|^3*b+b*(a|^3*b))+(a*(a|^2*b|^2) +b*(a|^2*b|^2))-(a*
  (a*b|^3)+b*(a*b|^3))+(a*b|^4+b*b|^4)
    .=(a|^5+b*a|^4)-(a|^4*b+b*b*a|^3)+((a|^2*a)*b|^2 +b*b|^2*a|^2)-(a*a*b|^3
  +b*b|^3*a)+(a*b|^4+b*b|^4) by POLYEQ_2:4
    .=(a|^5+b*a|^4)-(a|^4*b+b*b*a|^3)+(a|^2*a*b|^2 +b*b|^2*a|^2)-(a*a*b|^3+b
  |^4*a)+(a*b|^4+b*b|^4) by POLYEQ_2:4
    .=(a|^5+b*a|^4)-(a|^4*b+b*b*a|^3)+(a|^(2+1)*b|^2 +b|^2*b*a|^2)-(a*a*b|^3
  +b|^4*a)+(a*b|^4+b*b|^4) by NEWTON:6
    .=(a|^5+b*a|^4)-(a|^4*b+b*b*a|^3)+(a|^3*b|^2 +b|^(2+1)*a|^2)-(a*a*b|^3+b
  |^4*a)+(a*b|^4+b|^4*b) by NEWTON:6
    .=(a|^5+b*a|^4)-(a|^4*b+b*b*a|^3)+(a|^3*b|^2 +b|^3*a|^2)-(a*a*b|^3+b|^4*
  a)+(a*b|^4+b|^(4+1)) by NEWTON:6
    .=(a|^5+b*a|^4)-(a|^4*b+b|^1*b*a|^3)+(a|^3*b|^2 +b|^3*a|^2)-(a*a*b|^3+b
  |^4*a)+(a*b|^4+b|^(4+1))
    .=(a|^5+b*a|^4)-(a|^4*b+b|^1*b*a|^3)+(a|^3*b|^2 +b|^3*a|^2)-(a|^1*a*b|^3
  +b|^4*a)+(a*b|^4+b|^(4+1))
    .=(a|^5+b*a|^4)-(a|^4*b+b|^(1+1)*a|^3)+(a|^3*b|^2 +b|^3*a|^2)-(a|^1*a*b
  |^3+b|^4*a)+(a*b|^4+b|^5) by NEWTON:6
    .=(a|^5+a|^2*b|^3)-(a|^2*b|^3+a*b|^4)+(a*b|^4+b|^5) by NEWTON:6
    .=a|^5+b|^5;
  (a^2-a*b+b^2)*(a+b) =a^2*a+b*a^2-(a*(a*b)+b*(a*b))+(a*b^2+b*b^2+0*b^2)
    .=a|^ 3+b*a^2-(a*(a*b)+b*(a*b))+(a*b^2+b*b^2) by POLYEQ_2:4
    .=a|^ 3+b*a^2-(a^2*b+b*b*a)+(a*b^2+b|^ 3) by POLYEQ_2:4
    .=a|^3+b|^3;
  hence thesis by A1;
end;
