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 Th9:
  a1|^(m+1)+b1|^(m+1) = ((a1|^m+b1|^m)*(a1+b1) + (a1-b1)*(a1|^m-b1|^m))/2
  proof
    thus a1|^(m+1)+b1|^(m+1)
    = ((a1|^m+b1|^m)*(a1+b1) + (a1|^1-b1|^1)*(a1|^m-b1|^m))/2 by Th8
    .= ((a1|^m+b1|^m)*(a1+b1) + (a1-b1)*(a1|^m-b1|^m))/2;
  end;
