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
  a1|^(m+2) - b1|^(m+2) = (a1|^(m+1)+b1|^(m+1) + a1*b1*k)*(a1-b1) & a1*b1<>0
  implies a1|^m - b1|^m = (a1-b1)*k
  proof
    assume
    A1: a1|^(m+2) - b1|^(m+2) =
    (a1|^(m+1)+b1|^(m+1) + a1*b1*k)*(a1-b1) & a1*b1<>0; then
    A5: (k*(a1-b1)*(a1*b1))/(a1*b1) = k*(a1-b1) by XCMPLX_1:89;
    (a1|^(m+1)+b1|^(m+1))*(a1-b1) + a1*b1*(a1|^m-b1|^m) =
    (a1|^(m+1)+b1|^(m+1))*(a1-b1) + a1*b1*k*(a1-b1) by A1,Th21;
    hence thesis by A1,XCMPLX_1:89,A5;
  end;
