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

theorem
  for a,b be non zero Nat, n be odd Nat holds
    (a|^(n+2)+b|^(n+2))/(a+b) = a|^(n+1)+ b|^(n+1) - a*b*((a|^n+b|^n)/(a+b))
proof
  let a,b be non zero Nat, n be odd Nat;
  reconsider m = (n-1)/2 as Nat;
  a|^((2*m+1)+2) + b|^((2*m+1)+2) =
    (a+b)*(a|^((2*m+1)+1) + b|^((2*m+1)+1)) - a*b*(a|^(2*m+1)+ b|^(2*m+1))
      by RI3; then
  (a|^((2*m+1)+2) + b|^((2*m+1)+2))/(a+b) =
    (a+b)*(a|^((2*m+1)+1) + b|^((2*m+1)+1))/(a+b) -
      (a*b*(a|^(2*m+1)+ b|^(2*m+1)))/(a+b) by XCMPLX_1:120
  .= (a|^((2*m+1)+1) + b|^((2*m+1)+1)) -
    ((a*b)*(a|^(2*m+1)+ b|^(2*m+1)))/(a+b) by XCMPLX_1:89
  .= (a|^((2*m+1)+1) + b|^((2*m+1)+1)) -
    (a*b)*((a|^(2*m+1)+ b|^(2*m+1))/(a+b)) by XCMPLX_1:74;
  hence thesis;
end;
