reserve a,b for Complex;

theorem Th6:
  (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 Th5
    .=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;
