reserve a,b,c,d,x,j,k,l,m,n for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem Th8:
  ((a1|^m+b1|^m)*(a1|^n+b1|^n) + (a1|^n-b1|^n)*(a1|^m-b1|^m))/2
  = a1|^(m+n) + b1|^(m+n)
  proof
    thus ((a1|^m+b1|^m)*(a1|^n+b1|^n) + (a1|^n-b1|^n)*(a1|^m-b1|^m))/2 =
    (a1|^m*a1|^n + b1|^m*b1|^n + a1|^n*a1|^m + b1|^n*b1|^m)/2
    .=(a1|^(m+n) + b1|^m*b1|^n + a1|^n*a1|^m + b1|^n*b1|^m)/2 by NEWTON:8
    .=(a1|^(m+n) + b1|^(m+n) + a1|^n*a1|^m + b1|^n*b1|^m)/2 by NEWTON:8
    .=(a1|^(m+n) + b1|^(m+n) + a1|^(m+n) + b1|^n*b1|^m)/2 by NEWTON:8
    .=(a1|^(m+n) + b1|^(m+n) + a1|^(n+m) + b1|^(n+m))/2 by NEWTON:8
    .= a1|^(m+n)+b1|^(m+n);
  end;
