theorem
  1 <= k <= m implies (2*m+1)*x_r-seq(m).k = k*PI
  proof
    assume 1 <= k <= m;
    then
A1: x_r-seq(m).k = k*PI/(2*m+1) by Th19;
    (2*m+1)*(k*PI)/(2*m+1) = k*PI by XCMPLX_1:89;
    hence thesis by A1;
  end;
